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High-precision (p,t) reactions to determine reaction rates of explosive stellar processesMatić, Andrija

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High-precision (p,t) reactions to determinereaction rates of explosive stellar processes

Andrija Matic

RIJKSUNIVERSITEIT GRONINGEN

High-precision (p,t) reactions to determine reaction

rates of explosive stellar processes

Proefschrift

ter verkrijging van het doctoraat in deWiskunde en Natuurwetenschappenaan de Rijksuniversiteit Groningen

op gezag van deRector Magnificus, dr. F. Zwarts,in het openbaar te verdedigen op

maandag 11 juni 2007om 16.15 uur

door

Andrija Maticgeboren op 2 december 1974

te Valjevo, Serbia

Promotor:Prof. dr. M.N. HarakehCopromotor:Dr. A.M. van den Berg

Beoordelingscommissie:Prof. dr. G. P. A. BergProf. dr. K. HatanakaProf. dr. C. E. Rolfs

Contents

1 Introduction 11.1 Explosive stellar scenarios . . . . . . . . . . . . . . . . . . . . . . .. . . 1

1.1.1 Supernovae type II . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.2 Novae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.3 Supernovae type Ia . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.4 X-ray bursts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 CNO cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 NeNa and MgAl cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Astrophysical importance of22Na and26Al . . . . . . . . . . . . . . . . . 7

1.4.1 22Na . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4.2 26Al . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.5 Outline of this work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Theoretical model of stellar reaction rates 112.1 Stellar reaction rates . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 112.2 Non-resonant reaction rates (direct reaction rates) . .. . . . . . . . . . . . 122.3 Resonance reaction rates through narrow resonances . . .. . . . . . . . . 172.4 Reactions through broad resonances . . . . . . . . . . . . . . . . .. . . . 202.5 The astrophysically relevant excitation-energy regions for22Mg and26Si . . 22

3 Experimental setup and method 273.1 Experimental area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2 The GR spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.3 Matching between beam line and spectrometer . . . . . . . . . .. . . . . . 303.4 Over-focus mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.5 The detection and data-acquisition systems . . . . . . . . . .. . . . . . . 363.6 Experimental settings for the (p,t) reaction . . . . . . . . .. . . . . . . . . 38

i

Contents

4 Calibration 434.1 Software corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . .434.2 Background subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . .. 454.3 Reference data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.4 Fits of the spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.5 Calibration function . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 534.6 Calibration of26Si spectra . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5 22Mg data and their astrophysical implications 615.1 The24Mg(p,t)22Mg angular distributions . . . . . . . . . . . . . . . . . . 615.2 22Mg and its mirror nucleus22Ne . . . . . . . . . . . . . . . . . . . . . . 635.3 Calibration region (g.s. - 5.711 MeV) . . . . . . . . . . . . . . . .. . . . 645.4 Region above the proton-emission threshold (5.5042 MeV- 8.142 MeV) . . 655.5 Region above theα-emission threshold (8.142 MeV-10.5 MeV) . . . . . . 745.6 Region above 10.5 MeV . . . . . . . . . . . . . . . . . . . . . . . . . . . 795.7 Astrophysical implications for the18Ne(α,p)21Na reaction . . . . . . . . . 835.8 Astrophysical implications for the21Na(p,γ)22Mg reaction . . . . . . . . . 915.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

6 26Si data and their astrophysical implications 996.1 26Si and its mirror nucleus26Mg . . . . . . . . . . . . . . . . . . . . . . . 996.2 Calibration region (g.s. - 5.5123 MeV) . . . . . . . . . . . . . . .. . . . . 1016.3 Region above the proton-emission threshold (5.5123 MeV- 9.164 MeV) . . 1046.4 Region above theα-emission threshold (9.164 MeV) . . . . . . . . . . . . 1116.5 Astrophysical implications for the25Al(p,γ)26Si reaction . . . . . . . . . . 1116.6 Astrophysical implications for the22Mg(α,p)25Al reaction . . . . . . . . . 1166.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

7 Summary and conclusion 123

A DWBA and CC calculations 127

Nederlandse samenvatting en vooruitblik 131

Acknowledgment 135

Bibliography 137

ii

Chapter 1

Introduction

1.1 Explosive stellar scenarios

One of the goals of nuclear astrophysics is to explain the energy production in stars, andrelated to this, creation and abundance of the chemical elements. Since the discovery in1952 by Merrill [1] of radioactive technetium in spectra of S-stars, it became clear thatnuclear synthesis of heavier nuclei in a star’s interior does happen indeed. Furthermore,since technetium has a half-life of 105 to 106 years, which is a very short time period on acosmological scale, this implies that nuclear synthesis also takes place in our present times.

According to the Big-Bang theory matter created shortly after the Big Bang consistspredominantly of1H and4He. The heaviest created progenitor nucleus produced in signif-icant abundance is7Li, whereas all heavier nuclei are created in stars; see Ref.[2]. Theso-called main-sequence stars, which encompass the majority of stars, fuse hydrogen intohelium. After the exhaustion of hydrogen in the core of a star, the core will contract be-cause of the gravitational force and consequently the temperature and the pressure in thecore will rise. Under these conditions the helium burning process will start and carbon(C), and oxygen (O), will be produced. After the exhaustion of helium in the core, its con-traction will resume, leading to the ignition of C and O burning processes through whichheavier elements are created.

22Na and26Al are two long-living nuclei which eventuallyβ+ decay followed by sub-sequentγ-ray emission in the respective daughter nuclei. In case of explosive-burningscenarios, these nuclei can be ejected into the inter-stellar medium, and their subsequentγ-ray radiation can be detected with existingγ-ray observatories in outer space. Therefore,in this section we will discuss 4 explosive-burning scenarios for stars; these are novae, su-pernovae type I, supernovae type II, and X-ray bursts. Some nuclear processes which takeplace during the phases of explosive burning and which lead to the creation of heavier ele-ments than C and O will be discussed as well. These explosive scenarios have to be takeninto account in order to explain the isotopic abundances observed throughout the universe.

2 1. Introduction

1.1.1 Supernovae type II

A typical star for which one can apply the supernova II model has a mass of more than about8 M⊙ (mass of the Sun). These stars evolve much faster than most main-sequence stars,which have typical lifetimes of about 108 years. During its lifetime a star goes throughmany burning stages, starting with the cycles of hydrogen burning up to silicon burning,subsequently leading to the formation of a Fe-Ni core. Because the iron nuclei are mosttightly bound, a star cannot produce energy by the fusion of nuclei using iron or nickel asseed. Endothermic nuclear reactions will continue at extremely high temperatures of about4 T9 (T9=109 K). These reactions areγ-ray photo-disintegration of nuclei and electroncapture on protons leading to the production of neutrons andescaping neutrinos. The stellariron core loses energy through these two reactions, and consequently the pressure of thedegenerate electrons in the core decreases. Subsequently,the core contracts under thegravitational force and the temperature increases by compression. This dynamical collapseis starting at a density of about 109 g cm−3 and goes on to nearly nuclear matter density(1014 g cm−3). At that moment, the density in the core can not be increasedmuch furtherand the core collapse is stopped abruptly (in the order of a few milliseconds) leading to thecollapsing material bouncing on the core.

The bounce of the stellar material is very hard. Consequently, this leads to an outwardmoving compression wave which has such a high speed that it creates a shock wave. Ifthe shock wave is strong enough material can get such a high energy, leading to a velocitybeyond that of the escape velocity. In this case a large part of the stellar mass is ejectedand this is the supernovae phenomenon. If the mass of the progenitor star is between 8-20M⊙, the stellar core will evolve to a neutron star. If the progenitor mass is above 20 M⊙,the stellar core evolves to a black hole.

1.1.2 Novae

In case of a so-called nova, one is dealing with a binary system. One of the stars is a whitedwarf and the other one is a star near the main sequence or an aging star such as red giant.The white dwarf can be a star mainly consisting of carbon-oxygen (CO) or oxygen-neon-magnesium (ONeMg), formed after the helium-burning (He) orcarbon-burning stages, re-spectively.

In low-mass main-sequence stars hot gaseous matter can be described by the ideal gaslaw. In these circumstances a velocity distribution of electrons and nuclei can be describedby the Maxwell-Boltzmann velocity distribution. In case a star has consumed its nuclearfuel, it will collapse under the gravitational force with a huge increase in density. Underan enormous matter density the electron energy distribution is changed due to the Pauliexclusion principle. As the stellar material shrinks, its volume decreases and the number ofstates in a unit energy interval is reduced. Consequently, there will be less quantum states at

1.1. Explosive stellar scenarios 3

lower energies than is necessary for the Maxwell-Boltzmanndistribution to apply and moreelectrons remain at higher energy than would be expected from Maxwell-Boltzmann. Asthe density continues to increase more and more electrons will remain with higher energy,up to the moment when almost all states up to the Fermi level are filled. At this momentthe electrons form a degenerate gas which is under a huge pressure because of the electronsrapid motion. The high pressure provided by the electrons prevents further compression ofthe stellar material. In case of a white dwarf the electrons are pressed so tightly that furthercompression is not possible.

The Roche lobe is a space around a star in a binary system whichcontains all materialbound to that star. If a star expands beyond its Roche lobe, material can fall onto theother member of the binary system. In case of a nova the latteris a white dwarf anda hydrogen-rich accretion envelope can be created around it. This results from materialfalling from the star near the main sequence onto its accompanying star, the white dwarf.Since more and more hydrogen-rich material is accreted on top of the white dwarf, thetemperature of this envelope will rise up to the moment that the hydrogen burning processstarts. This temperature is around 0.02 T9. Because electrons are in a degenerate state, gasmaterial can not expand or cool before the degeneracy is lifted, and hence a rapid rise oftemperature under constant pressure and density will occur. Degeneracy is lifted when thelocal temperature reaches the Fermi temperature, causing arapid expansion of the envelopematerial into the interstellar medium. The typical nova peak temperature is 0.2 - 0.3 T9.

During a nova explosion the burning of hydrogen proceeds through the CNO cycles (seesection 1.2). Under these circumstances, high temperatures can be achieved and the CNOcycles can be broken, leading to rapid proton-capture (rp) reactions. These rp reactions canproduce nuclei with a mass higher than those taking part in the CNO cycles.

1.1.3 Supernovae type Ia

The model of a supernova type Ia is a binary system of a CO whitedwarf and a companionstar. Various types of companion stars have been assumed in different models. In themost common scenario, the white dwarf accretes material from a companion star up tothe moment that its core mass reaches the Chandrasekhar mass[3]. At that moment thegravitational force is strong enough to overcome the pressure from the degenerate electrongas, leading to a collapse of the star. At these high pressures and temperatures carbon andoxygen ignite in the core of the white dwarf and the burning front propagates outwards.

During a supernova explosion, a sequence of runaway nuclearreactions occur in thecore of the star. This is in contrast to regular nova explosions, where a thermonuclearrunaway happens at the bottom of the accreted envelope. Furthermore, a supernova typeIa explosion releases a much greater amount of energy than a typical nova explosion. It isbelieved that a supernova type Ia is not an important contributor to nucleosynthesis beyondiron.

4 1. Introduction

1.1.4 X-ray bursts

The standard model of an X-ray burst is based on a close binarysystem, where one memberof this system is a neutron star and the other one a hydrogen-rich star. Neutrons in a neutronstar are so densely packed that they form a degenerate gas. Because of the high density andthe deep gravitational well of a neutron star, a freely falling proton will arrive at the surfaceof the neutron star with an energy greater than 100 MeV [4]. Atthe surface, the high-energyproton will lose its energy via emission of many X-rays, mostly with an energy range below20 keV.

At the surface of the neutron star, the accumulated materialbecomes degenerate like innovae. Under high temperature hydrogen starts to burn via the hot CNO cycle (see section1.2). This process is rapid and occurs under a constant pressure up to the moment whendegeneracy of the neutron gas is lifted and it expands. When the cooling rate becomesequal to the energy production, X-ray bursters reach a peak-surface temperature of 2.5 T9.At the end, most of the initial helium and most of the other isotopes are converted intoheavy isotopes with mass heavier than A=72, up to100Sn, see Ref. [5]. In these processesproton-rich isotopes beyond56Fe can be produced.

1.2 CNO cycles

The cold CNO cycle is the fusion process of 4 hydrogen nuclei into helium with an en-ergy production of 26.73 MeV per cycle. This process takes place in hot (above 0.16 T9),hydrogen-rich environment where small amounts of heavier elements (C, N, O) act as cat-alysts, and thus their relative abundances remain unchanged during the process.

The cold and hot CNO cycles are presented in Fig. 1.1. The coldCNO cycle operatesat temperatures below 0.2 T9 and it is governed by the slowest reactions. These are theβ+

decays of13N and15O. When the stellar temperature reaches 0.2 T9, proton capture on13Nis more probable thanβ+ decay and the hot CNO cycle becomes operational. When thestellar temperature increases above 0.4 T9, an additional hot CNO cycle becomes availablevia the (α,p) reaction on14O.

At stellar temperatures beyond 0.5 T9 and 0.8 T9, the15O(α,γ)19Ne and18Ne(α,p)21Na reactions become possible, respectively. These two reactions provide abreak out from the CNO cycle into the NeNa cycle. Davidset al. [6] pointed out that thereis no significant break out from the CNO cycle via the15O(α,γ)19Ne reaction.

The18Ne(α,p)21Na reaction proceeds, at the temperatures required for explosive burn-ing of hydrogen in novae and X-ray bursts, through individual resonances above theα-emission threshold in the compound nucleus22Mg. Therefore, to calculate the rate for thisreaction, one has to know the properties of these high-lyingresonances.

1.3. NeNa and MgAl cycles 5

12C 13N

13C

14N15O

15N

12C 15N

15O14N14O

17F 18Ne 18F

19Ne21Na

Cold CNO Cycle

The Hot CNO Cycle

(p,γ)

(α,γ)(p,α)

(α,p)

β+

0.2 T9

0.4 T9

0.8 T9 0.5 T9

Figure 1.1: Cold and hot CNO cycles. Different nuclear reactions are marked with different arrowtypes.

1.3 NeNa and MgAl cycles

In hot stellar environments, where temperatures are higherthan those for the stable CNOcycles, other cycles become operational where nuclei with amass heavier than oxygenstart to act as a catalyst. These are the NeNa and MgAl cycles,which form a sequence aspresented in Fig. 1.2. These two cycles do not contribute significantly to the stellar energyproduction, because of the higher Coulomb barriers involved in the reactions presented inFig. 1.2. But they are important for the production of elements between20Ne and27Al.

22Na and26Al are two long-lived nuclei with a half-life of 2.602 years and 7.2×105

years, respectively. Their decay is followed byγ-ray emission at 1.275 MeV and 1.809MeV, respectively; see Fig. 1.3. Therefore, if these two isotopes can survive nova andX-ray burst explosions, their characteristicγ-ray emissions can be observed after the novaexplosion. Weiss and Truran [7] concluded that a nova can be an important source of26Alin the Galaxy, and that some nearby novae can produce amountsof 22Na which can bedetected withγ-ray observatories.

The detection of the 1.275 MeV and 1.809 MeVγ-rays will provide an excellent bench-

6 1. Introduction

21Ne 22Na

21Na 22Ne

20Ne 23Na 24Mg

25Al

25Mg 26Al

26Mg

27Al

27Si

(p,γ)

β+

(p,α)

NeNa cycle MgAl cycle

Figure 1.2: NeNa and MgAl cycles. Different nuclear reactions are marked with different arrowtypes.

26Alg.s.

26Mgg.s.

0+

2+

5+

β+

t1/2=7.2x105 yr

Eγ=1.809 MeV

3+

2+

0+

22Nag.s.

22Neg.s.

β+

t1/2=2.602 yr

Eγ=1.275 MeV

Figure 1.3: 26Alg.s. and22Nag.s. β+ decay followed byγ-ray emission.

mark for existing novae models. The 1.809 MeVγ-ray, associated with theβ+ decay of26Al nucleus, has been observed by the COMPTELγ-ray observatory based on board ofthe CGRO satellite [8]. The very same observatory could not detect any signal of the22Na1.275 MeVγ-ray [8]. However, the upper limit for the ejected amount of22Na from ob-served novae could be determined.

1.4. Astrophysical importance of 22Na and 26Al 7

1.4 Astrophysical importance of22Na and 26Al

1.4.1 22Na

The cold NeNa cycle is presented in the left panel of Fig. 1.2.From this figure we can seethat the production of22Na follows the

21Na(β+ν)21Ne(p,γ)22Nareaction path. The proton capture reaction on21Na becomes more probable than itsβ+-decay with rising stellar temperatures. Consequently, the

21Na(p,γ)22Mg(β+ν)22Nareaction path may lead to the production of22Na. Furthermore, proton capture can occur on22Mg (t1/2=3.87 s) at these temperatures. Due to the lowQ-value for the22Mg(p,γ)23Al re-action, the majority of23Al nuclei will, however, be destroyed via the inverse23Al(γ,p)22Mgreaction [9]. The lack of relevant data for the21Na(p,γ)22Mg reaction is one of the mainsources of uncertainty in calculations of the production of22Na. Therefore, the study of thenuclear structure of22Mg has been subject of many recent experiments [10, 11, 12, 13, 14,15, 16]. These studies together with the results from capture reactions using radioactivebeams of21Na [17, 18, 19] have contributed to a better understanding ofthe productionrate for22Na in stellar environments.

In Section 1.2 we mentioned that the18Ne(α,p)21Na reaction might provide a pos-sible break-out from the hot CNO cycle to the NeNa cycle. Thisreaction proceeds athigh temperatures, which can be found in explosive stellar environments, through individ-ual resonances above theα-emission threshold in the compound nucleus. Therefore, the18Ne(α,p)21Na reaction and the nuclear structure of22Mg have been the subject of severalexperimental investigations. Caggianoet al. [13] and Chenet al. [12] measured excitationenergies in22Mg above theα-emission threshold with an accuracy between 15 keV and 45keV. Bradfield-Smithet al. [20] and Groombridgeet al. [21] measured resonance strengthsand the excitation energies of levels in22Mg above 10 MeV with an accuracy of 50 keVand 140 keV, respectively. However, the accuracy achieved in these 4 experiments for thedetermination of the excitation energy is not satisfactoryfor the astrophysical rate calcula-tions. The errors in the excitation energies produce more than a 50% error in the calculated18Ne(α,p)21Na reaction rates. In the present work we measured excitation energies of lev-els in22Mg with an error less than 5 keV for levels below 10.5 MeV and approximately 20keV for levels above 10.5 MeV.

1.4.2 26Al

The 1.809 MeVγ-ray line was observed by the HEAO-C satelliteγ-ray observatory in1984 [22]. The discovery of26Al in the interstellar medium [23] demonstrated that indeednucleosynthesis processes are ongoing in our present time.The half-life of 7.2×105 yr for

8 1. Introduction

26Al is short compared to the time scale of the galactic chemical evolution. The production

25Al

26Si

25Mg

26Alg.s.

5+

0+ t1/2=6.4 s Ex=0.229 MeV

(p,γ)

(β+)

26Mgg.s.

0+

2+ t1/2=476 fs Ex=1.809 MeV

0+

5/2+

5/2+

Figure 1.4: 26Alg.s. and26Alm production andβ+ decay schemes. Different nuclear reactions aremarked with different arrow types.

mechanism for26Al depends on the stellar environment, the most promising productionsites being Wolf-Rayet stars, AGB stars [24], novae, and supernovae explosions. The two26Al production mechanisms are presented in Fig. 1.4. The firstone can occur at “lower”stellar temperatures above 0.035 T9, where26Al is produced via the reaction sequence:

25Al(β+ν)25Mg(p,γ)26Alg.s.(β+ν)26Mg∗(γ)26Mgg.s..

The half-life of the ground state of26Al is 7.2×105 yr. Its decay to the first-excited 2+

state in26Mg is followed byγ-ray emission of 1.809 MeV. However, if proton capture on25Al is faster than25Al β+ decay (which will occur at stellar temperatures higher than0.4T9), the reaction will follow the second path;

25Al(p,γ)26Si(β+ν)26Al∗(β+ν)26Mgg.s..

In this path26Si decays into the isomeric, first-excited 0+ state of26Al with a half-lifeof only 6.4 s, which subsequently decays to26Mgg.s.. Following this path there will be no1.809 MeVγ-ray emission, and galactic production of26Al cannot be observed. Ward andFowler [25] showed that for temperatures lower than 0.4 T9 there is no thermal equilib-rium between26Alg.s. and the first-excited state and consequently, they can be treated likeseparate species.

The25Al(p,γ)26Si reaction has not been measured directly, yet. However, the nuclearstructure of26Si has been the subject of several recent experimental studies [26, 27, 28,

1.5. Outline of this work 9

29]. In this context we also performed the28Si(p,t)26Si measurements above the proton-emission threshold with an energy resolution of 13 keV.

At stellar temperatures above 0.3 T9, energy production and nucleosynthesis in an ex-plosive hydrogen-burning environment are determined by the rp-process andαp-process.The proton-capture reaction rates are orders of magnitude faster thanβ-decay rates in therp-process. The reaction path consists of a series of (p,γ) reactions up to a nucleus wherefurther proton capture is inhibited. The proton-capture reactions can be inhibited by thenegative proton-captureQ-value, which is followed by proton decay, or by a small positiveproton-captureQ-value, which is followed by a photodisintegration process. The reactionflow has to wait for the slow nucleusβ-decay and that nucleus is denoted as a “waitingpoint”.

The 22Mg, 26Si, 30S, and34Ar isotopes areβ+ unstable and possible waiting points.These isotopes are important because most of the reaction flow is passing through them.Because of the small (p,γ) Q-value for 22Mg, photodisintegration prevents a significantflow through a subsequent23Al(p,γ) reaction. However, this waiting point can be bridgedby the22Mg(α,p)25Al reaction. This reaction is part of a chain

14O(α,p)(p,γ)18Ne(α,p)(p,γ)22Mg(α,p)(p,γ)26Si(α,p)(p,γ)30S(α,p)(p,γ)34Ar(α,p)(p,γ)38Ca

and has been theoretically investigated by Fiskeret al. [30]. These calculations wereperformed to explain bolometrically double-peaked type I X-ray bursts, see Refs. [31,32, 33]. The22Mg(α,p)25Al reaction has not been experimentally investigated previously,because there are no26Si data above theα-emission threshold which is located at 9.164MeV. We performed the28Si(p,t)26Si measurements and for the first time observed thenuclear structure above theα-emission threshold in26Si.

1.5 Outline of this work

In Chapter 2 the formalisms for the calculation of the nuclear reaction rates are given. Theexperimental set up used for our (p,t) reaction studies is explained in Chapter 3. The mo-mentum calibration for the obtained spectra is presented inChapter 4, whereas the analysisof the data obtained for22Mg and26Si are explained in Chapters 5 and 6, respectively. InChapter 7, the conclusions and an outlook of this work are given.

Chapter 2

Theoretical model of stellar reaction rates

In this chapter we will discuss the basic theoretical model necessary to calculate stellarreaction rates. The complete formalism and the notation aretaken from Ref. [4]. In thefollowing natural units will be used. Here, we will discuss only reaction rates induced by acharged particle.

2.1 Stellar reaction rates

The non-degenerate stars have a “simple” structure. The hottest and most dense region isthe core. Going from the interior outwards, the stellar temperature and pressure drop. Anormal non-degenerate star consists of a plasma containingnuclei and free electrons. Theplasma in the stellar core is fully ionized, and consists of various isotopes. Because thisplasma is in thermodynamic equilibrium, the velocity distributions of the nuclei can bedescribed by Maxwell-Boltzmann velocity distributions, see Ref. [4],

φ(υ) = 4πυ2( m

2πkT

)3/2

exp

(

−mυ2

2kT

)

(2.1)

This equation can be written in terms of the kinetic energy ofthe nucleus:

φ(E) ∝ E exp

(

− E

kT

)

(2.2)

where the most probable value of the kinetic energy is equal to kT .

The total reaction rate for a nuclear reactiona + B −→ c + D can be written as:

R = NaNB〈συ〉(1 + δaB)−1 (2.3)

whereNa andNB are the numbers of particles of typea andB per cubic centimeter in astellar plasma, respectively. The factor(1+δaB) prevents double counting in case identicalparticles interact with each other. Here,

〈συ〉 =

∫ ∞

0

φ(υ)υσ(υ)dυ (2.4)

12 2. Theoretical model of stellar reaction rates

is referred to as the reaction rate per particle pair, whereσ(υ) is the nuclear cross section,andφ(υ)dυ is the probability that the relative velocityυ between the particles involvedin the nuclear reaction is betweenυ andυ+dυ. The integration for exothermic reactionsextends fromυ=0 up to infinity, and for endothermic reactions the integration starts at thethreshold velocity for a particular reaction.

As mentioned before, for nuclear reactions taking place in the stellar interior, the dis-tribution of the relative velocityv of the interacting nucleia andB is described by theMaxwell-Boltzmann velocity distribution, and the reaction rate per particle pair is givenas:

〈συ〉 =

∫ ∞

0

∫ ∞

0

φ(υa)φ(υB)υaυBσ(υ)dυadυB. (2.5)

After a proper kinematic transformation, using the reducedmassµ and total massM , andintegration over the center-of-mass velocity (the cross section depends only on the relativevelocity between the interacting particles) we obtain the following equation:

〈συ〉 =

(

8

πµ

)1/21

(kT )3/2

∫ ∞

0

σ(E)E exp

(

− E

kT

)

dE (2.6)

2.2 Non-resonant reaction rates (direct reaction rates)

A nucleus is a positively charged entity, and therefore, twocolliding nuclei repel each otherwith a force proportional to the product of their respectivenuclear charges. This repulsiveforce leads, in the case of a charged particle undergoing an attractive nuclear interaction,to a potential barrier called the Coulomb barrier. For the fusion of two charged particlesthey have to penetrate through the Coulomb and centrifugal barriers; see Fig. 2.1. The totalrepulsive potential is given

V =ZBZae2

r+

l(l + 1)~2

2µr2(2.7)

whereZB andZa are the nuclear charges of the interacting particles,r is their mutualdistance,µ is the reduced mass of the projectile-target system, andl is the orbital angularmomentum.

The typical Coulomb barrier for the interaction between twolight nuclei is of the orderof a few hundred keV or higher. At a temperature of 0.0015 T9, which is typical for astar like the Sun, energies of the nuclei are of the order of 1 keV. However, in case ofthe highest predicted supernovae temperature of 9 T9, the corresponding energies for thenuclei are of the order of a few hundred keV. Therefore, the typical particle energies in thestellar environment are smaller than the repulsive Coulombpotential between the nuclei.However, Gamow [34] showed that nuclei can penetrate the barriers with a small but finiteprobability via the quantum-tunneling effect. The penetrability Pl through the Coulomb

2.2. Non-resonant reaction rates (direct reaction rates) 13

Figure 2.1: Schematic view of the combined nuclear, Coulomb, and centrifugal potentials. WhereRC (E) is a classical turning point for the Coulomb and centrifugal barriers. Figure taken from Ref.[4].

and centrifugal barriers is energy dependent and can be expressed as:

Pl(E, Rn) =1

F 2l (E, Rn) + G2

l (E, Rn)(2.8)

whereFl andGl are, respectively, the regular and the irregular solutionsfor the Coulombwave function for a given relative angular momentum, at the nuclear interaction radius of:

Rn = 1.35 × (A1/3a + A

1/3B ) fm (2.9)

whereAa andAB are the masses of the projectile and target, respectively, given in atomicmass units.

Non-resonant reactions are reactions with a one-step process, where a direct transitioninto a bound state occurs. Radiative capture, presented in Fig. 2.2, is one example of anon-resonant reaction. Other possible non-resonant reactions are: pickup and strippingreactions, Coulomb excitation, and charge-exchange processes.


a+B

Ecm

D

Radiative capture B(a,γ

Q

γ

)D

Figure 2.2: Radiative-capture reaction as an example for a non-resonant reaction.

In case of non-resonant reactions, the cross section is proportional to a single matrixelement. In our example for radiative capture it is given as:

σγ ∝| 〈D | Hγ | B + a〉 |2 (2.10)

whereHγ is an electromagnetic operator describing the transition.At a relative kineticenergy much smaller than the Coulomb barrier and for an orbital angular momentuml=0,the tunneling probability can be approximated as [4]:

P = e−2πη (2.11)

The quantityη is called the Sommerfeld parameter and it is equal to

η =ZBZae2

ℏυ(2.12)

The interaction cross section is dependent on the penetrability and the de Broglie wavelength, which describes the geometrical effects of the cross sectionσ ∝ πλ2 ∝ πk−2.Including all these contributions, we can write the cross section as:

σ(E) =1

ES(E)e−2πη (2.13)

The factore−2πη describes the penetration through the Coulomb barrier of point-like nuclei

2.2. Non-resonant reaction rates (direct reaction rates) 15

(lin.

sca

le)

S(E

)−F

AC

TO

R(lo

g. s

cale

)C

RO

SS

SE

CT

ION

Figure 2.3: An example of the cross section and the astrophysical S-factor for a charged-particlenon-resonant nuclear reaction. Figure taken from Ref. [4].

without orbital angular momentum (s-waves). The functionS(E) is called the astrophysi-cal S-factor and contains all the nuclear physics of the reaction.

In non-resonant reactions, the cross section varies continuously as a function of the en-ergy, see Fig. 2.3. For astrophysical applications we are usually interested in a cross sectionat an incident energy of a few keV where data usually do not exist. The astrophysical S-factor functionS(E) varies smoothly as a function of the energy for non-resonantreactionsin the region well below the Coulomb barrier, see lower panelin Fig 2.3. Because of thischaracteristic feature, the astrophysical S-factor is used to extrapolate measured cross sec-tions to energies relevant for the astrophysical environment. By combining Eqs. 2.13 and2.6 we obtain the equation for the reaction rate for non-resonant stellar nuclear reactions

〈συ〉 =

(

8

πµ

)1/21

(kT )3/2

∫ ∞

0

S(E) exp

(

− E

kT− b

E1/2

)

dE (2.14)

where the quantityb arises from barrier penetrability and is given as:

b = (2µ)1/2πe2ZBZa/~. (2.15)

Since for non-resonant reactions, the astrophysical S-factor varies slowly, the strongestinfluence on the reaction rate is caused by the exponential penetrability term (− b

E1/2 ) and


Figure 2.4: The Gamow peak is indicated with the shaded area. Figure taken from Ref. [4].

the exponential Maxwell-Boltzmann term (− EkT ). The exponential factor which is related

to the penetrability through the Coulomb barrier shifts theeffective distribution of the reac-tion rates to a higher energy E0. The convolution of these two exponential functions resultsin a peak of the integrand near the energy E0, which is usually much larger thankT , knownas the Gamow peak; see Fig. 2.4.

For a given stellar temperatureT , nuclear reactions can take place in a relatively narrowregion of energies around E0. Because S(E) varies slowly as a function of the energy, it canbe approximated by a constant value over the Gamow peak:

S(E) = S(E0) = constant, (2.16)

and the reaction rate has the form:

〈συ〉 =

(

8

πµ

)1/21

(kT )3/2S(E0)

∫ ∞

0

exp

(

− E

kT− b

E1/2

)

dE. (2.17)

By taking the first derivative of this formula, the position of the Gamow peak for some

2.3. Resonance reaction rates through narrow resonances 17

a+B

E

D

Q

E Γ

γ

D

Resonant radiative capture B(a, )Dγ

rR

Figure 2.5: Resonant radiative capture as an example of resonant reactions. Q=ma+mB− mD isthe reactionQ-value. ER is the center-of-mass projectile energy needed to populatethe centroid of aresonance state.

nuclear reaction can be determined. It is given by the following formula:

E0 =

(

bkT

2

)2/3

= 0.122(Z2BZ2

aµ)1/3T2/39 MeV (2.18)

The width of the Gamow peak is given by:

∆E =4√3

√

E0kT = 0.237(Z2BZ2

aµ)1/6T5/69 MeV (2.19)

In Section 2.5 we will show the relevant parameters for the Gamow windows for the18Ne(α,p)21Na,21Na(p,γ)22Mg, 25Al(p,γ)26Si, and22Mg(α,p)25Al reactions discussed inthis work.

2.3 Resonance reaction rates through narrow resonances

A resonant process is a two-step process, in which an excitedstate Er of the compoundnucleus is formed that subsequently decays into lower-lying states. The resonant reactionshows a rapid variation of the cross section over a small energy range. An example of aresonant reaction is presented in Fig. 2.5.

The reaction cross section for resonant reactions is proportional to two matrix elements:

σγ ∝| 〈D | Hγ | Er〉 |2| 〈Er | HD | B + a〉 |2 . (2.20)

Where the matrix element involving the operatorHD describes the formation of the com-pound stateEr, and the second matrix element describes the subsequentγ-decay. Thisprocess can happen only if the energy of the entrance channelmatches closely with the


energy of the resonance involved:

ER + Q = Er. (2.21)

This process can occur for all excited states above the threshold energyQ, when ER

satisfies the condition given by Eq. 2.21. In addition, a resonant state can be formedvia a given reaction channel if selection rules are fulfilled(angular momentum and parityconservation laws).

Resonance phenomena often occur in physical systems. The cross section for a resonantreaction can be written in the form

σ(E) ∝ ΓaΓc

(E − ER)2 + (Γ/2)2(2.22)

using the analogy with a damped oscillator driven by an external force.Γa andΓc are thepartial widths of the entrance and exit channels, respectively, andΓ is the total width of theresonant state. The cross section is exactly given by the Breit-Wigner formula:

σBW (E) = πλ2 2J + 1

(2Ja + 1)(2JB + 1)(1 + δaB)

ΓaΓc

(E − ER)2 + (Γ/2)2(2.23)

Whereλ is the de Broglie wave length and J, Ja and JB are the spins of the resonant state,of the projectile, and of the target, respectively. The term

ω =2J + 1

(2Ja + 1)(2JB + 1)(1 + δaB) (2.24)

is known as the spin statistical factor, which can be obtained by summing over all finalstates and averaging over initial states. The summing over the final states reflects thatthe probability for a given process increases with an increasing number of available finalstates. Because in the entrance channel the colliding nuclei can have (2Ja+1) and (2JB+1)substates, the factor 1

(2Ja+1)(2JB+1) reflects the probability that in the entrance channel thenuclei are in one particular substate.

The criterion for narrow resonances is that the resonance width is much smaller thanthe resonance energy:Γ ≪ ER. Ref. [4] presents a quantitative criterion:

Γ

ER≪ 10%. (2.25)

An example of a narrow resonance is given in Fig. 2.6. Under this circumstance theMaxwell-Boltzmann function changes very little over the resonance region and the termE exp

(

− EkT

)

in Eq. 2.6 can be approximated byER exp(

−ER

kT

)

and taken outside theintegral in Eq. 2.6:

2.3. Resonance reaction rates through narrow resonances 19

Figure 2.6: Schematic view of the Maxwell-Boltzmann energy distribution at a given stellar temper-ature T together with the cross section for a narrow resonance. Figure taken from Ref. [4].

〈συ〉 =

(

8

πµ

)1/21

(kT )3/2ER exp

(

−ER

kT

)∫ ∞

0

σBW (E)dE (2.26)

For the integration of the Breit-Wigner cross-section yield for a narrow resonance, wecan neglect the energy dependence ofλ, Γ, Γa andΓc:

∫ ∞

0

σBW (E)dE = π2λ2 2J + 1

(2Ja + 1)(2JB + 1)(1 + δaB)

ΓaΓc

Γ(2.27)

where the product

ωγ = ωΓaΓc

Γ=

2J + 1

(2Ja + 1)(2JB + 1)(1 + δaB)

ΓaΓc

Γ(2.28)

is referred to as the strength of the resonance. Low-energy narrow resonances well belowthe Coulomb barrier will have a larger probability for decayby γ-rays as compared toparticle decay. For example, in case of the (p,γ) reaction we haveΓγ ≫ Γp andΓγ ≈ Γ.Consequently the resonance strength depends on the partialdecay width by proton emissionωγ = ω

ΓγΓp

Γ ≈ ωΓp. However, in case the resonance energy is well above the Coulombbarrier,Γp will be much larger thanΓγ , andΓγ will be the dominant factor in the resonancestrength.


The particle partial width can be calculated as shown in Ref.[9] as

Γparticle =3~

2

µR2n

PlC2Sparticle (2.29)

whereµ is the reduced mass of the interacting particle, Sparticle is its spectroscopic factor(in our case proton or alpha), C is the isospin Clebsch-Gordan coefficient, and Pl is thepenetrability through the Coulomb barrier and centrifugalbarrier for orbital-momentumlevaluated at an interaction radius Rn; see Eq. 2.9. To calculate the particle penetrabilitythrough the Coulomb and centrifugal barriers we used the code PENE [35].

In the case where several narrow resonances contribute to the reaction rates, their con-tributions are simply summed to obtain the reaction rates for a particular reaction;

〈συ〉 =

(

2π

µkT

)3/2

ℏ2∑

i

(ωγ)iexp

(

− Ei

kT

)

(2.30)

We obtained this formula by combining Eqs. 2.26, 2.27 and 2.28.Inserting all the relevant quantities, the reaction rate can be calculated by the formula:

NA〈συ〉 = 1.54 × 105(µT9)−3/2

∑

i

(ωγ)i × exp(−11.605Ei/T9) [cm3s−1mol−1]

(2.31)where the resonance strength(ωγ)i is in units of eV, and the resonance energy Ei is in unitsof MeV.

2.4 Reactions through broad resonances

Here, we consider resonances which are broader than the relevant energy window for agiven stellar temperature. According to the definition given in Ref. [4], a broad resonanceis a resonance where

Γ

ER≥ 10%. (2.32)

The cross sectionσ(E) extends over a large range of energies, and the dependence ofthe cross section on energy needs to be taken into account forthe calculation of the stellarreaction rate according to Eq. 2.6. The energy dependence ofthe cross section can becalculated as:

σ(E) = σRER

E

Γa(E)

Γa(ER)

Γc(E)

Γc(ER)

(ΓR/2)2

(E − ER)2 + Γ2(E)/4(2.33)

where the cross section and the total width are known at the resonance energy:σR =

σ(ER), ΓR = Γ(ER). Obviously, knowledge of the energy dependence of the partial

2.4. Reactions through broad resonances 21

widths is necessary for the calculation of the cross section. In general, charged particlesneed to penetrate Coulomb and centrifugal barriers. The particle partial width can be cal-culated by

Γl(E) =2~

Rn

(

2E

µ

)1/2

Pl(E, Rn)θ2l , (2.34)

where the quantityθ2l is called the reduced width of the nuclear state, which represents

the probability of finding the excited state in the configuration l. Usuallyθl is determinedexperimentally. The penetrability factor is given in Eq. 2.8. With an increasing orbitalangular momentuml, the centrifugal barrier becomes larger than the Coulomb barrier andΓl drops rapidly. As an example, the calculated partial protonwidths for different valuesof l for the16O+p 17F reaction are presented in Fig. 2.7.

(MeV)

(eV

)

Figure 2.7: The calculated partial proton widthΓl for the reaction channel16O+p17F as function

of proton energy for values of the orbital angular momentuml= 0 to 6~. Figure taken from Ref. [4].

The energy dependence ofΓγ is given as:

Γγ(Eγ) = αLE2L+1γ (2.35)

whereL is the multipolarity of the emittedγ-ray andαL is constant for each multipolarity


depending on the nuclear structure of the resonance. Still,the energy dependence of thepartialγ-widths is not as strong as for particle emission, because ofthe particle penetrabil-ity through the Coulomb barrier. Partialγ-widths are of the order of 1 eV or smaller. Incontrast, the particle widths can be very small well below the Coulomb barrier, and theycan be in the order of MeV above the Coulomb barrier.

If the resonance is near or above the Coulomb barrier, the partial particle width varieslittle over the resonance region (E=ER ± ΓR/2); see Fig. 2.7. In contrast, for a resonancewell below the Coulomb barrier, the partial particle width for the entrance channel variesvery rapidly. On the other hand, the partial particle width for the outgoing channel, in caseof (α,p) reactions discussed here, varies more slowly, because the emitted particle has anenergy, which is increased by the amount of theQ-value of the reaction. The cross sectionis not symmetric with respect to ER because it varies much more for energies below theresonance energy ER as compared to energies above the resonance energy.

2.5 The astrophysically relevant excitation-energy regionsfor 22Mg and 26Si

In the previous sections we discussed direct reactions and the simplest two resonant reac-tions, which are important for the calculation of the stellar reaction rates. The formalismgiven in the previous sections will be enough to calculate rates for the18Ne(α,p)21Na,21Na(p,γ)22Mg, 25Al(p,γ)26Si, and the22Mg(α,p)25Al reactions discussed in this thesis.The Gamow window concept is directly applicable in the case of direct reactions. How-ever, it is useful for resonant reactions to calculate the position of the Gamow window fora particular stellar temperature T and to see in which excitation-energy region a particularresonance will dominate.

In Section 1.2 we already mentioned that at a stellar temperature above 0.8 T9 the18Ne(α,p)21Na reaction becomes possible. This reaction is one of the possibilities to linkreactions between the hot CNO cycle and the NeNa cycle. This breakout from the hot-CNO cycle gives the energy trigger for the X-ray bursts. For the more accurate X-raybursts models it is therefore necessary to obtain more precise data for22Mg resonances forthe entire span of stellar temperatures up to 2.5 T9. On the right side of Fig. 2.8 we presentthe positions of the Gamow windows (peak position and width)for the 18Ne(α,p)21Nareaction at 0.8 T9 and 2.5 T9.

In Section 1.4.1 we discussed the astrophysical importanceof the 21Na(p,γ)22Mg re-action as a tool to check present novae models. Since the novae peak temperatures reachvalues of 0.4 T9, we are interested in22Mg levels up to 5.95 MeV. For the X-ray bursts weare interested in temperatures up to 2.5 T9. On the left side of Fig. 2.8 we show the Gamowwindow for this reaction for the temperature of 2.5 T9. From Fig. 2.8 it can be seen that weare interested in the22Mg nuclear structure from the proton-emission threshold upto 11.5

2.5. The astrophysically relevant excitation-energy regions for 22Mg and 26Si 23

MeV.In Section 1.4.2 we already mentioned that25Al(β+ν)25Mg(p,γ)26Alg.s. can occur at a

stellar temperature above 0.035 T9 and25Al(p,γ)26Si(β+ν)26Al∗ at higher stellar temper-atures. Because supernovae explosions can be possible sources for26Al production we areinterested in the25Al(p,γ)26Si reaction rates for temperatures up to the supernovae peaktemperature of 4 T9. On the left side in Fig. 2.9 we indicate Gamow windows at 2.5 T9 and4 T9 for the 25Al(p,γ)26Si reaction at the X-ray bursts and supernovae peak temperatures,respectively. From the same figure it can be seen that for thisreaction we are interested in26Si levels up to 8 MeV.

The 22Mg(α,p)25Al reaction was not previously studied. On the left side of Fig. 2.9Gamow windows for this reaction are shown for temperatures of 1 T9, for the X-ray burstspeak temperature 2.5 T9 and supernovae peak temperature 4 T9. From the same figure itcan be seen that for this reaction we are interested in26Si levels up to 13 MeV.


8.141 MeV

0.8 T9

2.5 T9

18Ne + α

10.272 MeV

12.665 MeV

5.504 MeV21Na + p

2.5 T9

22Mg

18Ne+α21Na+p

Figure 2.8: Relevant astrophysical windows for the18Ne(α,p)21Na (right) and21Na(p,γ)22Mg (left)reactions at temperatures of 0.8 T9 and 2.5 T9 for the18Ne(α,p)21Na reaction and at a temperatureof 2.5 T9 for the21Na(p,γ)22Mg reaction. The full horizontal lines indicate the thresholds for protonandα emission, respectively. The dashed lines indicate for illustrative purposes the positions of twoknown levels in22Mg.

2.5. The astrophysically relevant excitation-energy regions for 22Mg and 26Si 25

25Al + p

Si

5.512 MeV

22Mg + α

2.5 T9

1 T9

2.5 T9

25Al + p

9.164 MeV

4 T9

4 T9

13 MeV

22Mg + α

26

Figure 2.9: Relevant astrophysical windows for the22Mg(α,p)25Al (right) and 25Al(p,γ)26Si (left)reactions at temperatures of 1 T9, 2.5 T9 and 4 T9 for the 22Mg(α,p)25Mg reaction and at tem-peratures of 2.5 T9 and 4 T9 for the 25Al(p,γ)26Si reaction. The full horizontal lines indicate thethresholds for proton andα emission, respectively. The line at 13 MeV excitation energy is markedfor illustrative purposes.

Chapter 3

Experimental setup and method

The astrophysics motivation to study the nuclear structureof 22Mg and26Si is given inChapter 2. The (p,t) reaction was used for the following reasons: TheQ-values of the (p,t)reaction on24Mg and28Si are strongly negative in comparison to those on other stable tar-get nuclei, namely−21.197 MeV and−22.009 MeV, respectively. Only12C and16O have

similarQ-values. The outgoing triton has a very high mass-to-chargeratio (m

q=3) resulting

in a magnetic rigidity large compared to other reaction products. Therefore, depending onthe proton bombarding energy, mainly tritons pass through the magnetic spectrometer. Inthis experiment, in addition to the lines resulting from the(p,t) reaction only four otherlines were observed resulting from13C(p,d) and17O(p,d) reactions.

3.1 Experimental area

The experiments24Mg(p,t)22Mg and28Si(p,t)26Si were performed at the Research Centerfor Nuclear Physics (RCNP) of Osaka University using a 100 MeV proton beam from theRing Cyclotron. The proton beam impinged on a 0.82 mg/cm2 thick 24Mg target and a0.7 mg/cm2 thick 28Si target. Both targets were self-supporting but contained12C and16Oimpurities. For the identification and subtraction of events from these contaminants, a 1mg/cm2 thick 12C target and a 1 mg/cm2 thick Mylar target were used. Although theseimpurities were sources of unwanted background they provided important information forthe energy calibration. The Grand Raiden (GR) spectrometerwas used for the momen-tum analysis of the outgoing tritons, with its standard detector system consisting of two2-dimensional Multi-Wire Drift Chambers (MWDC) followed by three plastic scintillatorsfor timing and particle-identification purposes. The MWDC detectors allowed for the de-termination of the horizontal and vertical positions of thetritons and consequently theirangles of incidence at the focal plane (see for further details Section 3.5).

The layout of the experimental facility at RCNP is depicted in Figure 3.1. A primarybeam is extracted from the AVF Cyclotron and injected into the Ring Cyclotron (RC) forfurther acceleration. The protons of 100 MeV were transported through the West-South(WS) beam line leading to the target chamber of the GR spectrometer located in the Westexperimental hall.

28 3. Experimental setup and method

Figure 3.1: Floor plan of the experimental hall at RCNP. Figure is taken from Ref. [36].

The new “fully dispersion-matched” WS beam line of the RCNP facility was designedfor high-resolution spectroscopy experiments. In our casethe resolution was mainly limitedby the target thickness rather than the resolving power of the spectrometer; see Wakasaetal. [37].

3.2 The GR spectrometer

The GR spectrometer has been designed for high-resolution spectrometry measurementsat RCNP. A detailed description of the GR spectrometer can befound in Ref. [38]. Aschematic view of the layout of the GR spectrometer is shown in Fig. 3.2. It consists of twodipole magnets (D1 and D2), two quadrupole magnets (Q1 and Q2), one sextupole magnet(SX) and one multipole magnet (MP). The function of the sextupole magnet is to minimize

3.2. The GR spectrometer 29

Figure 3.2: Schematic view of the spectrometer. Figure taken from Ref. [39].

the second-order ion-optical aberrations, (x|θ2) and (x|φ2) and to keep the horizontal focal-plane tilt angle at 45. The multipole magnet can generate magnetic fields with quadrupole,sextupole, octupole and decapole components. Its functionis to compensate for other ion-optical aberrations of the system. For spin-physics studies the last DSR magnet is addedfor the measurement of the in-plane polarization transfer.This dipole was not excited inthe present experiment.

The spectrometer can momentum analyze particles with a maximum rigidity of Bρmax=5.4Tm and has a remarkably high momentum resolving power of

p/∆p =Dx

Mx · ∆x0= 37000 (3.1)

for a monochromatic beam spot with the size of∆x0=1 mm (Ref. [37]), where Mx isthe magnification andDx the dispersion of the spectrometer. The design parameters ofthe spectrometer are listed in Table 3.1. Thus, with the momentum dispersion and the


Table 3.1: Design parameters for the GR spectrometer taken from Refs. [37, 38].

Mean orbit radius 3 mTotal deflection angle 162

Angular range −4 to 90

Momentum dispersion (Dx=(x|δ)) 15.45 mMomentum resolutionp/∆p 37000Momentum range 5%Tilting angle of focal plane 45

Focal-plane length 1200 mmMaximum magnetic rigidity 5.4 TmMaximum field strength (D1,D2) 1.8 THorizontal magnification (Mx=(x|x)) −0.417Vertical magnification (My=(y|y)) 5.98Horizontal acceptance angle ±20 mradVertical acceptance angle ±70 mradMaximum solid angle 4.3 msrFlight path of central ray 20 m

horizontal magnification listed in Table 3.1, the momentum resolving power for∆x0=1mm isp/∆p=37000. The observed spectral resolution with the old (WN) beam line wassignificantly less, mainly due to the momentum spread of the incident beam. Becausethe GR spectrometer is characterized by a large momentum dispersionDx=15.45 m and amagnificationMx=−0.417, a very large momentum dispersion ofDtgt = −Dx/Mx ≈37m at the target position is required in order to achieve lateral dispersion-matching. Thenew WS beam line is designed to achieve this beam-line dispersionDtgt. Consequently,the beam spot has a large horizontal size leading to uncertainties in the determination ofthe scattering angle. These uncertainties are coming from different scattering angles, whichdepend on the position on the target. Therefore, angular dispersion-matching is requiredto be able to reconstruct from measured focal-plane parameters the scattering angle withgood angular resolution.

The new WS beam line is designed to fullfill all required matching conditions: focuscondition, lateral dispersion-matching, and also angulardispersion-matching. Details ofthe WS beam line is given in Ref. [37].

3.3 Matching between beam line and spectrometer

In this section we will use transport notation following papers by Fujitaet al. [40] andWakasaet al. [37]. A charged particle is transported from the exit of the cyclotron tothe target and, after scattering, through the spectrometerup to the focal plane where thedetector system is positioned. Particles at the exit of the RC have the following horizontalcoordinatesx0, θ0 andδ0, denoting the position, angle and momentum deviations fromthe central trajectory, respectively. A particle ray with these coordinates is transported

3.3. Matching between beam line and spectrometer 31

by the beam line, represented by the transport matrixB, to the target location. There thecoordinates are given byx1, θ1 andδ1. At the target, the coordinates are transformed bythe T, defined byT, K, C, Eqs. 3.4, 3.5 and 3.6 below tox2, θ2 andδ2, and through thespectrometer by the spectrometer matrixS to the coordinatesxfp, θfp andδfp in the focalplane.

Thexfp andθfp coordinates of a particle at the focal plane can be describedin firstorder with theB andS matrix elements andT (Ref. [37]):

xfp = (s11b11T + s12b21)x0

+(s11b12T + s12b22)θ0

+(s11b16T + s12b26 + s16C)δ0

+(s12 + s16K)Θ

+higher-order terms

(3.2)

θfp = (s21b11T + s22b21)x0

+(s21b12T + s22b22)θ0

+(s21b16T + s22b26 + s26C)δ0

+(s22 + s26K)Θ

+higher-order terms

(3.3)

Where the suffixes 1, 2 and 6 refer to thex, θ andδ parameters, respectively. The parameterT is called the target function

T = cos(α − ΦT )/cos(ΦT ). (3.4)

The parameterα is the angle between the central ray of the incident beam at the targetposition and the central ray in the spectrometer andΦT is the angle between the normaldirection to the target and the direction of the incident beam (see Fig. 3.3). The factorΘ= θ1 − θ2 has the meaning of an “effective” scattering angle, and indicates how much thescattering angle of the particle is different from the nominal scattering angleα (see Fig.3.3), whereθ1 is the angle of an incident proton particle relative to the central ray of thebeam, andθ2 is the angle of the outgoing triton particle relative toα. The factorK is the“first-order” kinematic factor given by:

K = (1/pout)(∂pout/∂α) (3.5)

The parameterC is the “dispersion-matching factor”

C = (pin/pout)(∂pout/∂pin) (3.6)


wherepout is the momentum of the outgoing particle at the target position, andpin is theincoming momentum of the particle at the target position. For elastic scatteringC=1.

Target

X2

X1

α

β

θ

θ

1

2

Particle ray 1

Particle ray 2

Central ray into spectrometer

.ΦT

Central ray of beam

Figure 3.3: Schematic representation of the scattering of one particlewith coordinates(x1, θ1)

relative to the central ray of the beam. The(x2, θ2) are coordinates of outgoing particles andα is thescattering angle of the central ray.

The true scattering angle is defined by:

β = α + (θ2 − θ1) (3.7)

The minimum size of the imagexfp in the focal plane and thus the best resolution isachieved when the coefficients ofx0, θ0 andδ0 in Eq. 3.2 are zero. For a good definitionof the scattering angleθfp, we require that the coefficientδ0 in Eq. 3.3 should be zero.

The process of adjusting the spectrometer to eliminate theΘ term in Eq. 3.2 (K=−s12

s16)

is called thekinematic correction. Proper kinematic correction is achieved when particleswith different θfp converge at the same locationxfp. By requiring in addition that thecoefficient ofδ0 in Eq. 3.2 and Eq. 3.3 is zero we obtain the following conditions for thelateral dispersion

b16 =s16

s11(1 + s11s26K − s21s16K)

C

T(3.8)

and for the angular dispersion

b26 = (s21s16 − s11s26)K, (3.9)

at the target location, needed to optimize the momentum resolution and the angular resolu-

3.3. Matching between beam line and spectrometer 33

Figure 3.4: Schematic view of ion trajectories under different matching conditions (a) achromaticbeam transportation, no matching; (b) lateral dispersion-matching Eq. 3.8, ambiguity of the angularinformation still remains; (c) lateral and angular dispersion-matching Eqs. 3.8 and 3.9 are achieved.Figure is taken from Ref. [40].

tion at the focal plane of the GR spectrometer. From Eqs. 3.8 and 3.9 it can be seen thatthe parametersb16 andb26 of the beam line should depend on the parameterss12 ands16

needed for the kinematic correction of the spectrometer.

The resolving power of the matched system is given by:

R = (1/2x0)(s16/Mov) (3.10)

whereMov = (s11b11T + s12b21) has the meaning of an overall magnification and if thematching conditions are satisfied it has the form:

Mov = (s11b11T − s16b21K) (3.11)

The matching conditions are schematically illustrated in Fig. 3.4. An achromatic beamimpinging on the target is illustrated in Fig. 3.4 (a). A particle with a momentum difference∆p from momentum of the central ray is dispersed by the spectrometer and the resolutionis limited by the beam momentum spread. Under lateral dispersion-matching, shown inFig. 3.4 (b), a particle with a momentum difference∆p from momentum of the central rayhits the target at a different position in such way that the beam’s dispersion is compensatedby the spectrometer’s dispersion. But a particle with a momentum difference∆p from thatof the central ray is crossing the focal plane under different anglesθfp, so that an angularmeasurement in the focal plane does not allow the determination of the scattering angleat the target. When both lateral and angular dispersion-matching conditions have beenachieved simultaneously, Fig. 3.4(c), the position and theangle of the particle measured atthe focal plane (xfp, θfp) do not depend on the beam dispersion∆p and the measured anglein the focal plane corresponds to a unique scattering angle within measurement limitations.


3.4 Over-focus mode

In order to determine the angular distribution we need to perform an angular measurementwith good resolution. Near 0 the scattering angle depends equally on both the verticaland horizontal angle components. Consequently, we need to achieve a good resolution inboth directions. The GR spectrometer has a small vertical-angle magnification of≈ 0.17.Because of this small magnification, an ejectile with a largevertical scattering angle at thetarget will have a small angle at the focal plane. Under theseconditions it is impossibleto achieve a vertical resolution at the target better than 12mrad. The seemingly simplemethod of achieving a better angle measurement by increasing the distance between thefirst and the second position measurement, does not lead to much improvement due tomultiple scattering in the first detector and the resulting poor position measurement in thesecond detector. Therefore, in order to improve the vertical-angle resolution, the ion-opticalproperties of the GR spectrometer in the vertical directionwere considered in the transfermatrix formalism:

yfp = (y|y)y2 + (y|φ)φ2 + (y|yx)y2x2

+(y|yθ)y2θ2 + (y|yδ)y2δ1

+(y|φx)φ2x2 + (y|φθ)φ2θ2

+(y|φδ)φ2δ1 + higher-order terms

(3.12)

wherey2 andφ2 are the vertical position and the outgoing angle, respectively, of a particleat the target (similarly forx2 andθ2 in the horizontal direction), and(y|y),(y|φ) etc. arematrix elements of the spectrometer matrix, that transforms coordinates from the target tothe focal-plane coordinates [39].

In Fig. 3.5 the vertical trajectories are depicted for threedifferent settings:

(a) Focus mode of GR: the term(y|φ) from Eq. 3.12 is equal to zero for the central ray(ρ=300 cm).

(b) Over-focus mode: the strength of the Q1 magnet is increased and(y|φ) > 0.

(c) Under-focus mode: the strength of the Q1 magnet is decreased and(y|φ) < 0.

Because of the finite(y|φ)φ2 term in Eq. 3.12 in theover-focus andunder-focus modes,particles with different outgoing anglesφ2 at the target are transported to different positionsyfp in the focal plane. Therefore, it is possible to determineφ2 values from the measuredvalues ofyfp. In order to calibrate theover-focus mode, the(y|φ) and(y|φθ) terms shouldbe known. All other terms in Eq. 3.12 are expected to make the relationship betweenyfp

andφ2 more ambiguous, and for a useful angle determination they need to be small. Thelargest ambiguity comes from the vertical beam spot size at the target because of a largevertical magnification(y|y) ∼ 6. The horizontal beam spot size also induces an ambiguity.

3.4. Over-focus mode 35

Figure 3.5: The vertical trajectories of the scattered particle with the vertical positiony2 = ±1 mmand the outgoing angleφ2 =± 0.46 mrad from the target: (a) in the normal point-to-pointfocus modeof GR, (b) in theover-focus mode, (c) in theunder-focus mode. Figure is taken from Ref [39].

The accurate determination ofφ2 from yfp measurements depends on how much we canincrease the value of the(y|φ) term by a proper adjustment of the strength of the Q1 magnet,and how small the ambiguities can be made in practice. A detailed description of the newion-optical mode called “off-focus mode” applied to the GR spectrometer at RCNP can befound in Ref. [39].

This experiment was performed in theover-focus mode. For the reconstruction of thecoordinates at the target position from the measured coordinates at the focal plane, calibra-tion measurements with a multi-hole aperture (sieve-slit)were performed.


MWDC1

MWDC2

1 PLASTIC (1 mm) PL1



st

nd

rd

Al1 (1 mm)

Al2 (10 mm)

Figure 3.6: Schematic view of the focal-plane detector system of GR spectrometer.

3.5 The detection and data-acquisition systems

For our (p,t) experiment we used the standard focal-plane detector system which exists atRCNP. This system consists of two 2-dimensional multi-wiredrift chambers (MWDC’s) lo-cated at the focal plane for the determination of the positions and the angles of the particles.These MWDC’s are followed by a set of three plastic scintillation detectors. In betweenthese plastic scintillator detectors we placed aluminum plates to prevent secondary elec-trons produced by one scintillator to hit another scintillator. Furthermore, the thickness ofthe first two plastic scintillators and the aluminum plates were chosen in such a way thatthe highest energy tritons produced in the (p,t) reaction are stopped before they reach thethird plastic scintillator detector. The third scintillator detector was therefore used as a vetodetector to eliminate the background from scattered protons with a larger range. The layoutof the focal-plane detector system is depicted in Fig. 3.6. The signals from the first twoscintillators were used for triggering of the system as wellas particle identification throughenergy-loss measurements in these detectors.

The specifications of the MWDC’s are given in Table 3.2. Each MWDC has two anode-wire planes (X and U) sandwiched between three cathode planes. The anode planes consistof sense wires and potential wires. Fig. 3.7 shows the schematic structure of the X-wireplane. Sense wires are mounted at distances of 6 mm. For the U-wire plane the pitch ofthe wire spacing is 4 mm. The purpose of the potential wires isto produce a more uniform

3.5. The detection and data-acquisition systems 37

Table 3.2: The specifications of the MWDC.

Wire configuration X(0 vertical), U(48.2)Active area 1150 mm× 120 mmNumber of sense wires 192 (X), 208 (U)Cathode-anode gap 10 mmAnode-wire spacing 2 mmSense-wire spacing 6 mm (X), 4 mm (U)Sense-wire thickness 20µm gold-plated tungsten wirePotential-wire thickness 50µm gold-plated beryllium-copper wireCathode 10µm carbon-aramid filmGas mixture Argon + Iso-butane + Iso-propyl-alcohol

(71.4%), (28.6%)Entrance and exit window 12.5µm aramid filmDistance between two MWDC’s 250 mmPre-amplifier LeCroy 2735DC

electric field between the anode plane and cathode plane. Theelectrons created during theionization of the detector gas by charged particles can onlyproduce an avalanche near thesense wires. The drift time of the electrons is measured in order to determine a charged-particle trajectory. In the example shown in Fig. 3.7, the particle trajectory is determinedby using the drift-time information from four wires. For theproper determination of thetrajectory it is necessary to have information from a minimum of three wires. By combiningthe information from the MWDC’s, it is possible to determinea trajectory more accurately.The gas used in the MWDC is a mixture of 71.4% argon and 26.6% iso-butane, and a smallamount of iso-propyl-alcohol. The iso-propyl-alcohol is added to reduce deterioration ofthe detector gas, which can lead, e.g., to polymerization ofthe gas on the wire surfaces.

As signal preamplifier and discriminator LeCroy 2735DC cards are directly connected(without any cables) to the MWDC’s. The ECL output signals from these 2735DC cardswere transferred to LeCroy 3377 TDC’s, which digitize the signals of each wire. Thescintillator signals were digitized by a LeCroy FERA and FERET system (see Fig. 3.8).After readout, the events were stored in high speed memory buffer modules (HMS’s) inthe VME crate, and copied to a VMIC5576 reflective memory module (Rm5576 in VMEcrate) by an MC68040 based CPU board with the OS/9 operating system. The data fromthe experimental room were transported via a fiber-optic link to another Rm5576 modulein the counting room, and analyzed with a SUN work station. Then, the data are transferredfrom the SUN work station to the IBM RS/6000 SP station via an FDDI line. Finally, thedata are stored in list mode on a large hard disk.


Figure 3.7: Schematic structure of an X-plane of the MWDC. A typical track of a charged particleis illustrated. Figure is taken from Ref. [41].

On−line analysisRecordingEvent Build

Figure 3.8: Schematic view of the data-acquisition system used in the (p,t) experiment. Figure istaken from Ref. [42].

3.6 Experimental settings for the (p,t) reaction

As mentioned in the introduction, the goal of this experiment was to obtain high-resolutionspectra at 0 and also at more backward angles. Because of these requirements we decidedto perform a measurement at 0 using an existing Faraday cup in the first dipole D1 of the

3.6. Experimental settings for the (p,t) reaction 39

Target

Figure 3.9: Schematic view of the first dipole magnet of Grand Raiden withexisting Faraday cups;figure by Berg [43].

GR spectrometer and at 8 and 17. For the latter two angles we used a Faraday cup insidethe target chamber. With the required experimental conditions at 0, none of the existingFaraday cups at their standard locations inside the dipole magnet was able to collect theproton beam. However, the existing (3He,t) Faraday cup (see Fig. 3.9) could be used in ourexperiment by slightly moving it downstream.

Nevertheless, we still had a huge background at 0, caused by the proton beam. Thisbackground could be reduced significantly by changing the angle of the GR spectrometerto −0.3, still covering the angles around 0. In order to ensure the correct measurementof the integrated charge of the beam in this Faraday cup, it was checked against measure-ments in the other beam-stop inside the target chamber. In our data, we still had a tritonbackground from reactions on the Faraday cup, but we were able to remove it by using theinformation from the time of flight of the particles. Using this setup, the spectra in the vetoscintillator showed that we did not have any protons that were able to pass through the GRspectrometer. Therefore, we did not include the veto scintillator in the trigger logic.

We used beam intensities of 20 nA and 50 nA, for the measurements at−0.3 and the


backward angles, respectively. The smaller current at−0.3 is related to the (p,t) reactionwhen the Faraday cup inside of the D1 magnet of the GR spectrometer was used, whichintroduced a high counting rate in the detectors. The live time of the system varied between82% and 98% during this experiment.

Table 3.3: Angular and magnetic-field settings of the GR spectrometer.

Angle [] B1 [T] B2 [T] B3 [T]−0.3 0.71846 0.693008 0.71846 0.69300 0.6800017 0.71846 0.69300 0.68000

Three magnetic-field settings were used to cover the whole excitation-energy regionof interest with partly overlapping spectra. Table 3.3 lists the angular and the magnetic-field settings. Table 3.4 shows the excitation-energy regions of interest for the24Mg and28Si targets covered by the different magnetic fields. A spectrum will be indicated by thetarget (24Mg, 28Si, carbon or Mylar), the GR angle (−0.3, 8 or 17) and the magnetic-field settings (B1, B2, B3). As an example, the overlapping spectra for the24Mg(p,t)22Mgreaction at a scattering angle of 8 are shown in Fig. 3.10.

Table 3.4: The excitation-energy range for the different magnetic-field settings for24Mg and28Sitargets, respectively

target B1 B2 B324Mg -0.5 up to 6.5 MeV 5.5 up to 12.0 MeV 7.5 up to 14.0 MeV28Si -1.3 up to 5.7 MeV 4.7 up to 11.2 MeV 6.8 up to 13.2 MeV

3.6. Experimental settings for the (p,t) reaction 41

excitattion energy [MeV]

excitation energy [MeV]

excitation energy [MeV]

coun

ts/2

keV

coun

ts/2

keV

coun

ts/2

keV

B1

B2

B3

Mg(p,t) Mg excitation energy at GR angle 8o2224

Excitation energy (MeV)



Cou

nts/

2 ke

VC

ount

s/2

keV

Cou

nts/

2 ke

V

24 22Mg Mg(p,t) Θ=8 ο

Figure 3.10: 24Mg(p,t)22Mg excitation-energy spectra obtained at three magnetic-field settings anda scattering angle of 8. Overlapping areas can be seen in this figure.

Chapter 4

Calibration

As explained in Chapter 1, the22Mg and26Si levels above the proton-emission threshold(5.5042 MeV and 5.512 MeV, respectively) up to an excitationenergy of about 13 MeVare of astrophysical interest. We performed measurements covering the excitation-energyregion from the ground state up to 13 MeV, and used the low-lying well-known levels of22Mg for momentum calibration of the spectra. The reaction24Mg(p,t)22Mg was chosenfor the momentum calibration because the uncertainty of themass of22Mg (0.27 keV) issmaller than for26Si (1 keV), and there are more accurate energy levels available for 22Mg.

4.1 Software corrections

Owing to the kinematics of the reactions, particles emerging from the target at the sameexcitation energy of the residual nucleus but at different reaction angles have differentmomenta. This, together with the optical aberrations, can influence the resolution if theangular acceptance is large enough. The plots on the left side of Fig. 4.1 show the effectsof kinematics and aberrations on the energy spectra of our experiment.

In order to achieve the best possible resolution for our (p,t) spectra we performed soft-ware corrections to compensate for these effects. Preceding this, sieve-slit measurementshave been made to correlate target coordinates with focal-plane coordinates; see Ref. [44].In the analysis we used a procedure outlined by Yosoi [45]. The correction parameters aredetermined by using distributions of the horizontal projection,θfp, and vertical projection,φfp, of the scattering angle,Θscatt, measured in the focal plane versus the position,xfp, inthe focal plane:θfp versusxfp andφfp versusxfp. The left panel of Fig. 4.1(a) shows the2-dimensional spectrumθfp versusxfp. Every curved vertical line corresponds to a differ-ent discrete state. In order to remove these curvatures a fourth-order polynomial functionwas employed. The right panel of Fig. 4.1(a) shows the same spectrum after correctionsand straight vertical lines can be observed. Similarly the left panel of Fig. 4.1(b) showscorrelations ofφfp versusxfp. In order to correct these effects a second-order polynomialfunction was employed. The right panel of Fig. 4.1(b) shows the same spectrum aftercorrections.

The effect of these corrections on the resolution for our (p,t) spectra is illustrated inFig. 4.1(c) where we show the24Mg(p,t)22Mg position spectra before and after perform-

44 4. Calibration

ing the corrections for the−0.3 scattering angle. Because the kinematic effects are largerfor the larger scattering angles, the effects of these corrections are also much larger atthese backward angles. In case of full angular acceptance (−1.2 < θfp <1.2 and−3.2 < φfp <3.2), we fitted the ground state of22Mg for the uncorrected and thecorrected spectra obtained at−0.3 scattering angle with a Gaussian function. The ob-tained resolutions (FWHM) were 43 keV and 19 keV for the uncorrected and the correctedspectra, respectively. From the difference in the achievedresolution, it can be seen that thesoftware corrections employed for the kinematics and the aberrations are crucial to achievehigh resolution spectra.

However, by using these corrections we were not able to achieve fully straightθfp

versusxfp andφfp versusxfp spectral lines, for the whole angular acceptance along the

CorrectedUncorrected

x−position (mm)

x−position (mm) x−position (mm)

x−position (mm)

x−position (mm)

(a)

(b)

(c)

used

reg

ion

used

reg

ion

x−position (mm)

Cou

nts/

mm

θ fp

(deg

)(d

eg)

fpφ

Figure 4.1: (a) and (b): Plots of the horizontal scattering angleθfp and the vertical scattering an-gle φfp versus the position in the focal plane before and after corrections. (c) The24Mg(p,t)22Mgposition spectrum before and after corrections measured with the spectrometer at−0.3.

4.2. Background subtraction 45

x−position at focal plane (mm)

backgroundT

ime

of fl

ight

(ch

anne

ls)

Figure 4.2: The two-dimensional time-of-flight versus x-position spectrum for the measurementat−0.3 and the B2 magnetic-field setting. The background due to tritons from the Faraday cup isremoved by setting a gate as shown. The characteristic levelstructure for the24Mg(p,t)22Mg reactionis visible in the lower part of this 2-dimensional display.

entire focal plane. It can be seen from Fig. 4.1 that still substantial aberrations occur for−0.6 > θfp andθfp > 0.4 and|φfp| > 1.5. Therefore, we made a restriction on thedata in the angular region of−0.6 < θfp < 0.4 and−1.5 < φfp < 1.5, indicated in theright panels of Figs. 4.1(a) and 4.1(b), respectively. Thisimproved the energy resolution(FWHM) of the ground state to 13 keV.

4.2 Background subtraction

In Section 3.6 we mentioned that at the spectrometer angle of−0.3 we had a huge back-ground originating from the (p,t) reaction on the Faraday cup inside the GR spectrometer.In the two-dimensional spectrum of the time-of-flight versusxfp it is easy to identify whichpart is the background (see Fig. 4.2), and by a simple cut on the time-of-flight spectra weremoved this background.

As mentioned before we used self-supporting targets of24Mg and28Si with a thicknessof 0.82 mg/cm2 and 0.70 mg/cm2, respectively. Both targets had impurities of12C and16O, which have larger (p,t) cross sections and similar reactionQ-values compared to24Mg

46 4. Calibration

Og.

s.14

10C

g.s.

14O

10C

6.59

0 2

3.35

4 2+ +

24Mg(p,t) 22Mg

Position at focal plane (mm)

Cou

nts/

0.2

5 m

m

B1Θ=−0.3ο

Figure 4.3: The 22Mg spectrum taken at−0.3 and the magnetic-field setting B1; only stronglypopulated impurity lines are indicated with their known excitation energies.

and28Si. Therefore, our spectra are affected by impurity lines from the12C(p,t)10C and16O(p,t)14O reactions (see Fig. 4.3).

24Mg(p,t) 22Mg


Cou

nts/

2.5

keV

Cou

nts/

2.5

keV 14O 6.590 2 +

t

B2Θ=−0.3 ο

Figure 4.4: Part of the22Mg spectrum taken at the magnetic-field setting B2 before andafter sub-traction of the14O contaminant line. Due to “evaporation” of the impurities during the experiment,the ratio between the22Mg lines and contaminant lines was changing, as can be seen bycomparingthe 6.590 MeV14O level in Figs. 4.3 and 4.4 taken at different times.

4.3. Reference data 47

By taking additional data on a Mylar and a natural carbon target, we could identify andsubtract impurity lines from the22Mg and26Si spectra. The first step in the subtractionmethod was to subtract the10C lines from the Mylar spectra after normalization using iso-lated peaks. In this way, we obtained a pure14O spectrum without10C peaks or continuum.Subsequently, we subtracted the normalized10C and14O impurity spectra from the22Mgand26Si spectra. To determine the normalization coefficients forthe subtraction methodwe used the well-separated and the strongly populated impurity levels at low excitation en-ergy (the10C g.s. and14O g.s. in Fig. 4.3). Fig. 4.4 shows the part of the24Mg(p,t)22Mgspectrum withEx ≈ 6.1 - 6.5 MeV before and after the subtraction of the14O impurityline at 6.590 MeV.

In case of the spectra obtained at the B3 magnetic-field settings, which cover the higherexcitation-energy regions, lines originating from the13C(p,d)12C and the17O(p,d)16O re-actions are used to perform background subtraction. The (p,d) lines were easy to removefrom our spectra by using the∆E/E particle identification in the first and second plasticscintillator detectors PL1 and PL2.

Because of the highest statistics and the smallest influenceof kinematics, the excitationenergies were determined from the measurements at−0.3. In case where an impurity lineis covering a level in the triton spectrum taken at−0.3, the excitation energy for that levelwas determined from the spectra at 8 or 17.

4.3 Reference data

The first magnetic-field setting (B1) covers the energy rangein 22Mg approximately from−0.5 MeV up to 6.8 MeV. Fig. 4.5 represents the24Mg(p,t)22Mg spectrum obtained withB1=0.71846 T at an angle of−0.3. Because of its high statistics, this spectrum was cho-sen for the momentum calibration. For the calibration of Bρ/Bρcentral only 24Mg(p,t)22Mglines were used. Their excitation energies were taken from Ref. [16]. We used the mostaccurate energy levels below the proton-emission threshold that were published by Sew-eryniaket al. [16], who studied the high resolution21Na(p,γ)22Mg reaction. We used sixstrongly populated calibration lines (see Fig. 4.5) indicated by their excitation energies.These lines are distributed approximately from−460 mm up to 360 mm along the focalplane, which is active from−600 mm to 600 mm in total.

The levels from Ref. [16] used for the Bρ/Bρcentral calibration are indicated with thesuperscript∗ in the second column in Table 4.1. The third column in this table shows thedata by Batemanet al. [11]. The levels indicated with the superscripta are from Endt[46], and were used by Batemanet al. [11] for their calibration. Differences exist be-tween the following values of Refs. [16] and [11]: (4.4020(3) - 4.3998(42), 5.7110(10)- 5.7139(12)), although the first two values agree within theerror bars. It is important to

48 4. Calibration

mention that Batemanet al. [11] used only two previously known22Mg levels at 5.0370MeV and 5.7139 MeV for the calibration of their (p,t) spectra. But the 5.7139 MeV levelis in discrepancy with the corresponding level listed in Ref. [16] and the 5.037 MeV levelhas a three times larger error than the corresponding level listed in Ref. [16]. In order tocorrect Bateman’s energy calibration a simple linear function was employed. The correctedenergies are shown in the fourth column of Table 4.1. The new calibration gives a betteragreement between the excitation energies of Batemanet al. [11] and Seweryniaket al.[16]. These energies will be compared with our results rather than those from Ref. [11].

g.s.

3.30

8

5.03

5

5.71

1

Peak position [mm]

C g.s.

Og.s.

10

14

4.40

21.24

7

24Mg(p,t)

B1Θ=−0.3

Ep=98.7 MeV

ο

22Mg

Cou

nt/0

.25

mm

Peak position (mm)

Figure 4.5: The 24Mg(p,t)22Mg spectrum obtained with magnetic-field setting B1 at an angle of−0.3. Note: Lines used in the calibration are marked with their excitation energies in MeV. Theexcitation energies given in this figure are rounded off to three digits beyond the comma.

4.4. Fits of the spectra 49

Table 4.1: Energy levels of22Mg from the literature used in the present analysis. All energies aregiven in MeV. Data from Seweryniaket al. [16] are compared with those from Batemanet al. [11],and from Caggianoet al. [13]

Jπ Seweryniak Bateman Bateman Caggianoet al. [16] et al. [11] et al. [11] et al. [13]21Na(p,γ)22Mg 24Mg(p,t)22Mg corrected 25Mg(3He,6He)22Mg

0+ g.s.∗ - - g.s.a

2+ 1.24718(3)∗ - - 1.2463a

4+ 3.30821(6)∗ - - 3.3082a

2+ 4.4020(3)∗ 4.3998(42) 4.4013(42) 4.4009a

2+ 5.0354(5)∗ 5.0370(14)a 5.0362(14) 5.033(7)(1+) 5.0893(8) 5.0897(17) 5.0887(17) 5.094(6)4+ 5.2931(14) 5.2957(16) 5.2939(16) 5.301(4)2− 5.2960(4) - - -3+ 5.4524(4) 5.4543(16) 5.4519(16) 5451(5)2+ 5.7110(10)∗ 5.7139(12)a 5.7106(12) 5.7139a

0+ - 5.9619(25) 5.9577(25) -- - 6.0458(30) 6.041(30) 6.051(4)4+ - 6.2464(51) 6.241(51) 6.246(4)6+ 6.2542(3) 6.2464(51) 6.241(51) 6.246(4)1+ - 6.3226(60) 6.3170(60) 6.329(6)2+ - 6.613(7) 6.606(7) 6.616(4)- - 6.787(14) 6.780(14) 6.771(5)

The levels indicated with∗ were used for the present calibration;Levels indicated witha are used by Batemanet al. and Caggianoet al. for theircalibrations.

4.4 Fits of the spectra

For the determination of the position of the peaks we used twoGaussian functions and asquare-hyperbola function as defined in Eq. 4.1. This combination gives a lower reducedχ2 compared to, for example, a pure Gaussian or Breit-Wigner shape. The parameterswere obtained by fitting the first-excited state of22Mg (1.247 MeV). For the region Ex <

5.5042 MeV this shape was fixed,i.e. the only free parameters were the peak positions andthe heights. This function was used for all states below the proton-emission threshold of5.5042 MeV, since the instrumental width of the peaks was larger than the natural widthof the corresponding levels. Fig. 4.6(a) shows the first-excited state of22Mg fitted by ourfunction 4.1 together with the difference between the data and the fitted function.

50 4. Calibration

Two Gaussian functions and square hyperbola:

f(x) = y0

exp(− (x−x0)2

x21

ln2) x < x0

exp(− (x−x0)2

x22

ln2) x0 < x ≤ x0 + η x2

A

x − x0+

B

(x − x0)2x > x0 + η x2

(4.1)

with:x0 Abscissa value of the maximum of the Gaussian function,y0 Function value atx0,x1 Half width at half maximum of the Gaussian function forx < x0,x2 Half width at half maximum of the Gaussian function forx > x0,A, B Parameters to insure that the model function is differentiable along the

abscissa,η Distance between the starting points of the square hyperbola andx0 in units

of x2 (absolute distance =η · x2).

For levels above the proton-emission threshold the fitting function was changed to aGaussian and two exponential functions (Eq. 4.2).

Gaussian function and two exponential functions:

f(x) = y0

exp(− (x−x0)2

x2 ln2) x0 − η1 x < x < x0 + η2 x

exp(A1(B1 + x)) x ≤ x0 − η1 x

exp(A2(B2 − x)) x ≥ x0 + η2 x

(4.2)

with:x0 Abscissa of the maximum of the Gaussian function,y0 Function value atx0,x Half width at half maximum of the Gaussian function,A1,2, B1,2 Parameters to insure that the model function is differentiable along the

abscissa,η1, η2 Distances between the starting points of the exponential functions andx0 in

units ofx (absolute distance =ηi · xi, i can be 1 or 2).

The shape of the first-excited level above the proton-emission threshold was used todetermineη1 andη2. This change was introduced to accommodate the natural width thatincreases above the proton threshold. This particular function allows to fix the tails (η1, η2)while the width (∆x) can be adjusted. Fig. 4.6(b) shows the first-excited state above theproton-emission threshold of22Mg fitted with our function 4.2. For the magnetic-fieldsetting B2 the function 4.2 was fitted to the 7.2183 MeV level in order to determineη1 andη2. This line was positioned close to the center of the focal plane and had good statistics(see Fig. 4.7). The same line shape is also used for the spectrum obtained at the magnetic

4.4. Fits of the spectra 51

(a) Two Gaussina functions plus a square hyperbola

1.247 MeV

Position (mm/4)

Position (mm/4)

Cou

nts

Cou

nts

5.711 MeV

(b) Gaussian function plus two exponential functions

Position (mm/4)

Cou

nts

Cou

nts

Position (mm/4)

Figure 4.6: Fit for 22Mg atΘlab=−0.3 (a) level atEx= 1.247 MeV by the function given in Eq. 4.1and (b) level atEx= 5.711 MeV by the function given in Eq. 4.2, together with residues of the fits.See Fig. 4.5.

field B3; since none of the peaks in this spectrum provides acceptable statistics to determinethe parametersη1 andη2 listed in Eq. 4.2.

After removing the background contributions caused by the (p,t) reaction on the Fara-day cup and by contaminants in the target, as explained above, no additional backgroundwas assumed to be present in the excitation-energy region below 7.2183 MeV. For higherenergies a constant background was assumed.

In the 22Mg spectra, we defined four regions of interest (see Table 4.2) motivated by

52 4. Calibration

7.21

8

6.22

5

6.22

5

6.03

6

Cou

nts/

2.5

keV

7.21

8

5.95

46.

036

B2

6.22

6

5.71

1


Θ=−0.3

24Mg(p,t) 22Mg

ο

Figure 4.7: The24Mg(p,t)22Mg spectrum at the magnetic field B2 and at−0.3. Note that the linesused in the calibration are indicated by their excitation energies. See Fig. 4.5.

astrophysical reasons and due to the calibration procedure. Because of the low statisticsand the increase in the natural width at higher excitation energies, we changed the binningfor these regions from about 1.75 keV/channel in region one to about 14 keV/channel inregion four as shown in Table 4.2.

Table 4.2: Regions of interest and binning in the spectra.

Name Energy Binning1 calibration g.s. - 5.711 MeV 0.25 mm2 above ofp-emission threshold 5.711 - 8.142 MeV 0.5 mm3 above ofα-emission threshold 8.142 - 10.5 MeV 1 mm4 above of 10.5 MeV 10.5 - 13.0 MeV 2 mm

4.5. Calibration function 53

4.5 Calibration function

A correction for the energy loss of tritons in the target is taken into account in the calibrationprocedure. Our assumption is that on average the24Mg(p,t)22Mg reaction takes place at thecenter of the24Mg target. Following this approximation, the triton energyloss is calculatedfor half of the target thickness. The data for this correction are obtained from the code“LISE++” written by Tarasovet al. [47].

The recent direct21Na(p,γ)22Mg measurements performed with a radioactive21Nabeam by the TRIUMF-ISAC group (Bishopet al. [17]) obtained 0.2057(5) MeV for theresonant energy of the first-excited level above the proton-emission threshold. On basisof the adopted excitation energy 5.714 MeV at that time, Bishop et al. [17] calculated anew value for the mass excess of22Mg. However, the most recent21Na(p,γ)22Mg mea-surements [16] show that the excitation energy of the first-excited level above the proton-emission threshold is 5.711(1) MeV. Thus, the mass excess for 22Mg as given by Bishopetal. [17] has been underestimated. Two new experiments have beenperformed by Mukher-jeeet al. [48] and Parikhet al. [49] to measure the mass of22Mg. The measurements donewith the ISOLTRAP Penning-trap mass spectrometer at CERN [48] yield the most accuratevalue for the22Mg mass excess of−399.92(27) keV. Because our statistical error is largerthan 0.5 keV for the relevant peak positions, we are not sensitive enough to use the error of0.27 keV for the absolute Bρ calibration. We deliberately increased the error up to a valueof 0.55 keV, so we achieved aχ2

reduced ≈1 for the calibration function. This 0.55 keV isused later in our analysis of the22Mg mass error.

Two types of errors are included in the fitting procedure usedto obtain the calibrationfunction: (a) one regarding the peak position and (b) the other related to the absolute Bρvalues. The error of the beam energy is not included, neitherin the fitting procedure, norin the calculation of the energy levels. In fact, a variationin the beam energy of±100 keVintroduces less than 0.1 keV difference in the resulting excitation energies.

Four impurity lines were used to determine the beam energy, namely the ground statesof 10C, 12C and16O populated through the12C(p,t)10C, 13C(p,d)12C (at 8 and 17) and17O(p,d)16O reactions, respectively. These impurity lines were foundat the magnetic-fieldsetting B3, which covers the higher excitation-energy region. They were extracted by usinga ∆E/E particle identification in the PL1 and PL2 detectors. Panel (a) of Fig. 4.8 showsthe spectrum obtained with the Mylar target at 8 and magnetic-field setting B3. Panel (b)shows the (p,d) spectra extracted using the above-mentioned particle-identification cut.

For the calibration of Bρ/Bρcentral as a function of the triton position in the focalplanexfp we used a quadratic polynomial function. The six strongly populated22Mglevels indicated by their excitation energy in Fig. 4.5 wereused for the calibration. Thecode LSH2000 [50] was used to obtain the parameters of the calibration function. The

54 4. Calibration

Position in focal plane (mm)

Position in focal plane (mm)

Cou

nts/

mm

Cou

nts/

mm

13C(p,d)

O(p,d)17

(p,d) spectrum

(a)

(b)

Mylar spectrum collected at B3 and oΘ=8

Figure 4.8: Panel (a) shows the Mylar spectrum without the PID for the deuterons. Panel (b) showsthe (p,d) spectrum obtained from the Mylar spectrum by usinga PID cut for the deuterons.

Table 4.3: Differences between experimentally determined and calculated absolute kinetic energiesfor outgoing deuterons and tritons at Ep=98.700 MeV.

Reaction Magnetic-field Scattering angle KINEMA Experiment Difference() (MeV) (MeV) (keV)

1 12C(p,t)10C B1 −0.3 73.1205 73.1165 4.02 13C(p,d)12C B3 8 94.5782 94.5808 −2.63 13C(p,d)12C B3 17 93.8093 93.8044 4.94 17O(p,d)16O B3 8 95.6897 95.6875 2.2

calibration function for the magnetic-field setting B1 andΘ=−0.3 is:

Bρ/Bρcentral = (A + B · xfp + C · x2fp)/(2.155389 T·m) (4.3)


• The parameterxfp is the position in mm along the focal plane.

• The parameter 2.155389 T·m is Bρcentral for the B1 magnetic-field setting.

• A=(2.155174± 0.000005)×103 T·m

• B=(−9.8272± 0.0015)×10−2 T·m/mm

• C=(−2.80± 0.05)×10−6 T·m/mm2

The obtained reducedχ2 in this procedure is 0.97. Note: The22Mg mass uncertainty isdeliberately increased up to a value of 0.55 keV.

The proton beam energy of 98.7 MeV was determined such that the discrepancies be-tween the experimentally determined kinetic energies for the outgoing deuterons and tritons(the sixth column in Table 4.3) and the energies calculated with the code KINEMA [51](the fifth column in Table 4.3) for the same beam energy are minimized. These discrepan-cies might be a consequence of kinematics (different peak shape) and energy loss causedby unknown deposition of carbon and oxygen in the target.

The levels measured at both magnetic-field settings B1 and B2and at−0.3 spectrom-eter angle are listed in Table 4.4. In the first column, the excitation energies determined atthe magnetic-field setting B1 are listed. These levels were used for the calibration of themagnetic-field setting B2. We used the same Bρ/Bρcentral function which was employedfor the−0.3, B1 setting (Eq. 4.3). For other magnetic-field settings theposition of all lev-els are shifted by a small valuedx, to lie on one calibration line. The valuedx is obtainedby the function:

Bρ/Bρ[central]B1= (A + B · (xfp − dx) + C · (xfp − dx)2)/(2.155389 T·m) (4.4)

where the coefficients A, B and C are the same as in Eq. 4.3,xfp is the position of the levelobtained by fitting the spectra,Bρ[central]B1

is the magnetic field B1. The only unknownvalue is the small shiftdx. The same procedure was used for the calibration of all otherspectra (B1, B2, and B3 for 8 and 17). The resulting energies, for these three levels,in the magnetic-field setting B2 are listed in the second column of Table 4.4. This showsthat we do have a consistent parameterization for the momentum calibration over the wholefocal plane. Adopted values for the last two levels are shownin the third column of Table4.4. The averaging process is performed only for those levels for which the excitationenergy is measured in the extrapolation area of the present calibration. Furthermore, theyare used for the calibration at magnetic-field setting B2 andat magnetic-spectrometer angle−0.3. This is performed in order to decrease a possible error in the extrapolation of thepresent calibration.

In the error of the excitation energy calculated from Eq. 4.3only the error for thexposition in the focal plane enters as statistical error. Thesystematic error for the excitation

56 4. Calibration

energy includes errors originating from the reaction-angle determination and from the massof 22Mg (0.55 keV), which are added quadratically. The systematic and statistical errorsare linearly added to obtain the total error, which is quotedfor the present22Mg data.

Table 4.4: The excitation energies obtained for three levels at the B1 and B2 magnetic-field settingsand their weighted averaged energies.

Level energy Level energy LevelB1 setting B2 setting averaged energy5.7100(13) 5.7085(11) 5.711(1)∗

5.9536(14) 5.9539(11) 5.9538(9)6.0352(14) 6.0366(11) 6.0361(9)

Level energy is given in MeV.∗ this is the calibration value taken from Ref. [16].

For the calibration of the B3 magnetic-field setting we used only two levels with thelargest statistics; one at 9.7520(24) MeV and the other at 10.2717(15) MeV (see Fig. 4.9).

Tables 4.5 and 4.6 list our results up to 7.2183(10) MeV together with reference data.Here, we will discuss the main features of the calibration. The detailed discussion of theobserved22Mg levels will be presented in the next chapter.

Table 4.5: Excitation energies in22Mg below 5.711 MeV (the proton-emission threshold is at 5.5042MeV).

Jπ Present Seweryniak Bateman Caggiano McDonald Chenwork et al. [16] corrected et al. [13] et al. [52] et al. [12]

0+ g.s.∗ g.s. - g.s.a g.s. g.s.a

2+ 1.24718∗ 1.24718(3) - 1.2463a 1.244(32) 1.2463a

4+ 3.30821∗ 3.30821(6) - 3.3082a 3.269(50) 3.3082a

2+ 4.4020∗ 4.4020(3) 4.4013(42) 4.4009a 4.378(35) 4.408(12)a

2+ 5.0354∗ 5.0354(5) 5.0362(14) 5.033(7) 5.032(30) 5.029(12)a

1+ (5.092(5)) 5.0893(8) 5.0887(17) - 5.130(35) -4+ 5.2947(23) 5.2931(14) 5.2939(16) 5.301(4) 5.286(30) 5.272(9)2− - 5.2960(4) - - - -3+ 5.454(4) 5.4524(4) 5.4519(16) 5.451(5) 5.433(25) -2+ 5.7110∗ 5.7110(10) 5.7106(12) 5.7139a 5.699(20) 5.711(13)a

Level energy is given in MeV.∗ Values from Ref. [16] used for calibration in the present work.a Used for calibration in the previous articles.

In Tables 4.5 and 4.6 the present data are compared with thoseof Seweryniaket al.[16], Bateman’s corrected data, Caggianoet al. [13], and McDonaldet al. [52]. A goodagreement can be observed between our experimental resultsand the those of Ref. [16].However, there remain differences between the present dataand previous ones for the levels


Θ=8Ex=98.700 MeV

Mg(p,t) Mg

o

Excitation energy [MeV]

coun

ts/ 1

0 ke

V

10.2

742

9.75

19

24 22

Θ=8o10.2

72

9.75

2

Cou

nts/

10

keV


24Mg(p,t) 22Mg

B3

Figure 4.9: The24Mg(p,t)22Mg spectrum measured at the magnetic field B3 and a scatteringangleof 8. The calibration lines are marked by their excitation energies in MeV. See Fig. 4.5.

at 6.0413(30) MeV, and 6.2411(51) MeV (see Bateman corrected data), above the proton-emission threshold. The confidence in our calibration is based on the agreement with thevalues from Refs. [16] and [52]. Values from Ref. [52] are in agreement with those fromRef. [16]. The calibrations of all other references are based on those of Ref. [46] and as wealready showed for Ref. [11], these authors used a differentenergy for the last level in theircalibration which introduced a discrepancy with Ref. [16].The quality of our calibrationis further substantiated by the agreement for the 6.7688(12) MeV level with all previousdata [11, 12, 13] and the level at 7.2183(10) MeV with the values from Refs. [14, 46, 52].Moreover, the same calibration is used for our28Si(p,t)26Si data and our measured data for26Si are in agreement with published ones.

58 4. Calibration

Table 4.6: Energy levels in22Mg above the proton-emission threshold (5.5042 - 7.220 MeV).

Jπ Present Seweryniak Bateman Caggiano McDonald Chen Bergwork et al. [16] corrected et al. [13] et al. [52] et al. [12] et al. [14]

2+ 5.7110∗ 5.7110(10) 5.7106(12) 5.7139 5.699(20) 5.711 -0+ 5.9538(8) - 5.9577(25) - 5.945(20) - -- 6.0361(8) - 6.0413(30) 6.051(4) - 6.041(11) 6.059(9)

6.2261(10) - 6.2411(51)a 6.246(4)a - - 6.244(9)a

(6+) - 6.2542(3) 6.2411(51)a 6.246(4)a 6.263(20) 6.255(10) 6.244(9)a

- 6.306(9) - 6.3170(60) 6.329(6) - - -- 6.578(7) - - - 6.573(20) - -- 6.602(9) - 6.606(7) 6.616(4) - 6.606(11) 6.606(9)3− 6.7688(12) - 6.780(14) 6.771(5) 6.770(20) 6.767(20) 6.766(12)3− 6.8760(12) - - 6.878(9) - 6.889(10) -- 7.027(9) - - - - - -- 7.045(7) - - - - - -- 7.060(7) - - - - - -- 7.079(9) - - - - - -0+ 7.2183(10) - - 7.206(6) 7.201(20) 7.169(11) 7.216(9)

Level energy is given in MeV.∗ Values from Ref. [16] used for calibration in the present work. a Unresolved doublet.

4.6 Calibration of 26Si spectra

For the28Si(p,t)26Si spectra we used the same calibration as for the24Mg(p,t)22Mg spectra;see Eq. 4.3. Because of the smaller uncertainty in the mass of22Mg as compared to26Si,the calibration with22Mg is more accurate, and resulted in smaller errors.

In Table 4.7 we list the reference data for26Si, that were used in previous and thepresent calibration of28Si(p,t)26Si spectra. We used five strongly populated calibrationlines (see Fig. 4.10) indicated by their excitation energies. The calibration was performedby shifting the calibration points to the calibration line,using the same procedure as em-ployed with Eq. 4.4. This procedure also provided an excellent test for the calibration thatwe performed for the24Mg(p,t)22Mg spectra.

For our analysis we decided to use the calibration without the levels at 5.145 MeVand 5.515 MeV. These values are obtained by Bardayanet al. [27], Caggianoet al. [28]and Parpottaset al. [29] by using the doublet at 4.806 MeV as the last calibrationpoint.In the present work we resolved the above-mentioned doublet(see discusion in Section6.2). Therefore, we decided not to use the levels at 5.145 MeVand 5.515 MeV in ourcalibration procedure in an attempt to examine their excitation energies without influenceof the above-mentioned doublet on the calibration.

In this way, we obtain an independent measurement for these levels. From the averageddata determined from the measurements made at−0.3 and magnetic-field settings B1 and

4.6. Calibration of 26Si spectra 59

Table 4.7: Jπ and excitation energies (in MeV) of26Si levels relevant for calibration.

Jπ Present work Parpottas Caggiano Bardayan Endtet al. [29] et al. [28] et al. [27] [46]

0+ g.s.∗ g.s.a g.s.a g.s.a g.s.2+ 1.7959(2)∗ 1.7959a 1.7959a 1.7959a 1.7959(2)2+ 2.7835(4)∗ 2.7835a 2.7835a 2.7835a 2.7835(4)0+ 3.3325(3)∗ 3.3325a - 3.3325a 3.3325(3)- (3.749(4) 3.756a - 3.756a 3.756(2)- - - - - 3.842(2)- - - - - (4.093(3))2+ 4.1366(28) 4.138(4) 4.144(8) 4.155(2) 4.138(1)3+ 4.186(4) 4.183(4) 4.211(16) 4.155(2) 4.183(11)2+ 4.446(3)∗ 4.446a 4.446a 4.445a 4.446(3)- 4.8057(25) 4.806a 4.806a 4.805a 4.806(2)- 4.8270(25) - - - -2+ 5.1415(14) 5.145(4) 5.140(10) 5.145(2) -- - - - - 5.229(12)4+ 5.286(6) 5.291(4) 5.291 5.291(3) 5.330(20)4+ 5,5116(25) 5.515(4) 5.526(8) 5.515(5) 5.562(28)

∗ Used in the present calibration.a Used for calibration in the previous articles.

B2 we obtain 5.1415(14) MeV and 5.5116(20) MeV which are in good agreement with thetabulated data 5.145(4) MeV and 5.515(4) of Ref. [29], respectively. For a more detaileddiscussion of the observed26Si levels see Section 6.2.

The calibration of28Si(p,t)26Si spectra is performed by using the calibration of24Mg(p,t)22Mg spectra as is explained above. Therefore, the calculated26Si excitation en-ergies are more sensitive for a possible error in the beam-energy determination. Changingthe value of the beam energy by 100 keV resulted in a change in the26Si excitation energiesby up to 1.5 keV for states at high excitation energy.

In the error of the26Si excitation energies calculated from Eq. 4.3 only the error in thex position in the focal plane enters as statistical error. Thesystematic error includes errorsoriginating from the reaction-angle determination, errorof the mass of26Si (1 keV) andthe error resulting from the uncertainty in the beam energy.These three systematic errorsare quadratically added to obtain the total systematic error. The systematic and statisticalerrors are linearly added to obtain the total error, and thiserror is quoted for the present26Si data.

For astrophysical reasons it is important to perform more accurate26Si mass measure-ments, especially as this may influence the value of the proton-emission threshold. This canbe decisive to tell whether the 5.515(4) level is above or below the proton-emission thresh-old. However, this level is well below the Gamow window for the 25Al(p,γ)26Si reaction

60 4. Calibration

28Si(p,t)

1.79

6*

g.s.

*

coun

ts/ 2

.5 k

eV


2.78

4*

4.44

6*

Θ=−0.3

Θ=8

θ=173.

332*

o

o

Cg.

s.10

o

Si26

Figure 4.10: The 28Si(p,t)26Si spectra obtained with magnetic-field setting B1 and spectrometerangles of−0.3, 8 and 17. The calibration lines are marked with their excitation energy in MeV.See further Fig. 4.5 for more details.

at the lower temperatures (above T9=0.01) where the X-ray burst ignites. Consequently,its contribution will not be taken into account. More information regarding this problem ispresented in Ref. [49]. These authors measured accurately the mass of26Si, showing thatthe mass of26Si is 6 keV heavier than what was tabulated in Ref. [53]. A direct conse-quence of this work is that the value of the26Si proton-emission threshold decreases from5.518 MeV to 5.5125 MeV.

Chapter 522Mg data and their astrophysical implications

In this chapter we will discuss the24Mg(p,t)22Mg data in detail. Our data will be comparedwith previous results and possible spin assignments will begiven. This information will beincluded in the18Ne(α,p)21Na and21Na(p,γ)22Mg reaction-rate calculations and used in amodel describing X-ray bursts and ONe (oxygen-neon) novae.

5.1 The24Mg(p,t)22Mg angular distributions

To obtain angular-distribution parameters for the (p,t) reaction, we performed measure-ments at three different GR spectrometer angles (−0.3, 8 and 17). Differential crosssections were calculated using the equation

dσ

dΩ= 2.66 · 10−4 At

D

nt

µ

Zp

Ω

1

Q(1 − τ)[mb/sr] (5.1)

where:

Zp is the elementary charge of the projectile.At is the mass of the target [g/mol].Ω is the effective solid angle [sr].nt is the number of counts inΩ.D is the detection efficiency (in our experiment above 82%).µ is the thickness of the target [mg/cm2].Q is the integrated charge [nC].τ is the dead time.

The deduced differential cross sections were compared withdistorted-wave Born ap-proximation (DWBA) calculations performed with the code DWUCK4 [54]. A coupled-channels (CC) calculation including inelastic scatteringin the entrance and exit channelsusing the code CHUCK3 [55] did not change appreciably the shape of the calculated angu-lar distributions. The proton optical-potential parameters for the input channel were takenfrom Ref. [56]. For the outgoing channel the parameterization of Ref. [57] was usedwhich is based on the analysis of the26Mg(3He,t)26Al reaction. All parameters used in theDWBA and CC calculations are given in Appendix A.

62 5. 22Mg data and their astrophysical implications

We found a large discrepancy between the measured and calculated angular distribu-tions for the first three22Mg levels (g.s. 0+, 1.247 MeV 2+ and 3.308 MeV 4+). Theangular distributions of these three levels are presented in Fig. 5.1 together with the an-gular distributions calculated with DWUCK4. In addition, we found a discrepancy in theobtained differential cross section for the12C(p,t)10C reaction compared to those from theprevious experiments [58, 59], where the observed angular distributions are less steep. Thereason for this discrepancy remains unknown.

Because of the discrepancy between the measured and calculated angular distributions,we performed mirror spin-parity assignments on basis of already known data for22Na. Forthe22Na levels for which the spin and parity are not known, we made assignments on basisof theoretical calculations given by Ref. [60]. Because the24Mg(p,t)22Mg reaction pro-ceeds mainly through a direct reaction mechanism it mainly populates states with naturalparity in 22Mg. Therefore, we assumed that the previously observed levels that have notbeen observed in our experiment are unnatural-parity or high-spin (natural-parity) states.

0 5 10 15 20Θ

c.m. (deg)

10-3

10-2

10-1

100

dσ/d

Ω (

mb/

sr)

g.s. 0+

calculated

0 5 10 15 20Θ

c.m. (deg)

1.247 2+

calculated

0 5 10 15 20Θ

c.m. (deg)

10-3

10-2

10-1

100

3.308 4+

calculated

Figure 5.1: The measured and calculated angular distributions for the g.s. 0+, 1.247 MeV 2+, and3.308 MeV 4+ levels of22Mg. See further Fig. 4.5 for more details. The solid line is used to guidethe eye.

5.2. 22Mg and its mirror nucleus 22Ne 63

5.2 22Mg and its mirror nucleus 22Ne

It can be seen from Eq. 2.30 that resonance strength data are necessary for the calcula-tion of the reaction rates. Up to now several experiments [17, 19, 18, 20, 21] have beenperformed with the aim to measure the resonance strengths for the 21Na(p,γ)22Mg and18Ne(α,p)21Na reactions directly. They succeeded in measuring the resonance strengthsfor eight resonances above the proton-emission threshold and eight resonances above thealpha-emission threshold. To investigate the rest of the22Mg levels and their influence onthe 21Na(p,γ)22Mg and18Ne(α,p)21Na reaction rates we need spin-parity values, and thepartial widthsΓp, Γγ , or Γα for these resonances.

The source for spin-parity assignments andΓγ andΓα data can be mirror levels in22Ne, which is a stable nucleus. Therefore, its nuclear structure is much better known than22Mg. The main data source for excitation energies and spin-parities of levels in22Ne isRef. [61]. Nevertheless, spin-parity values are not known for a number of levels, withinthe astrophysical region of interest (below 14 MeV). Consequently, additional spin-parityvalues have been taken from the theoretical calculations listed in Ref. [60]. These valuesare indicated by the superscriptT in our tables and figures.

The main obstacle in our spin-parity mirror assignments is the scarcity of definite spin-parity assignments in22Mg and the lack of it above the 0+ level at 7.218 MeV. Therefore,spin-parity assignments at higher energies become more andmore uncertain.

α-spectroscopic factors for22Mg were taken from Refs. [20, 21]. In addition, we as-sumed that corresponding22Mg and22Ne mirror states above the22Neα-emission thresh-old have the sameα-spectroscopic factor. For this purpose, we used theα-spectroscopicfactors for22Ne levels above the alpha-emission threshold that are listed in Ref. [62]; seealso Section 5.6.

For the21Na(p,γ)22Mg reaction-rate calculations we took for22Mg theγ-resonance-widths,Γγ , of corresponding22Ne mirror states and corrected these for the differences inγ-ray transition energies; see Section 5.8. In22Ne, theΓγ level widths for all levels below9.69 MeV are calculated from their life-times, where known,since for these levels onlyγ-decay is allowed.

Here, we explained why additional data for reaction-rate calculations are needed. Forthis purpose, spin-parity,Γγ andΓα data taken from22Ne will be used for mirror levels in22Mg. In this way, we attempt to minimize the errors introducedby parameters which arenot experimentally obtained for22Mg nuclei. The procedures for using these parameters inthe reaction-rate calculations are described in the following sections.


Table 5.1: The calibration region covering the22Mg excitation energies below 5.711 MeV (proton-emission threshold 5.512 MeV), with the adopted spin-parity assignments.

Jπ Present work Ref. [16] Ref. [11] Ref. [13] Ref. [52] Ref. [12] adopted(p,t) (p,γ) (p,t) (3He,6He) (3He,n) (16O,6He) present

0+ g.s. g.s. - g.s.a g.s. g.s.a g.s.2+ 1.24718∗ 1.24718(3) - 1.2463a 1.244(32) 1.2463a 1.24718(3)4+ 3.30821∗ 3.30821(6) - 3.3082a 3.269(50) 3.3082a 3.30821(6)2+ 4.4020∗ 4.4020(3) 4.4013(42) 4.4009a 4.378(35) 4.408(12)a 4.4020(29)2+ 5.0354∗ 5.0354(5) 5.0362(14) 5.033(7) 5.032(30) 5.029(12)a 5.0346(5)1+ (5.092(5)) 5.0893(8) 5.0887(17) - 5.130(35) - 5.0893(8)4+ 5.2947(23) 5.2931(14) 5.2939(16) 5.301(4) 5.286(30) 5.272(9) 5.2938(10)2− - 5.2960(4) - - - - 5.2960(4)3+ 5.4540(40) 5.4524(4) 5.4519(16) 5.451(5) 5.433(25) - 5.4524(4)2+ 5.7110∗ 5.7110(10) 5.7106(12) 5.7139a 5.699(20) 5.711(13)a 5.7101(5)

All energies are in MeV.∗ Levels from Ref. [16] used in our calibration.e Used for calibration in previous articles.Data in the fourth column are the corrected data of Batemanet al., which we alreadydiscussed in Section 4.3 Table (4.1).

5.3 Calibration region (g.s. - 5.711 MeV)

In this section we will discuss the unnatural-parity statesup to 5.711 MeV. The natural-parity states have already been discussed in Section 4.5 andwere used in the energy cal-ibration. In Fig. 5.2 spectra obtained at−0.3, 8 and 17 are shown with calibrationlevels indicated by *. It can be seen that in addition to the six calibration levels takenfrom Ref. [16], we observed three weakly excited levels at 5.092(5) MeV, 5.2947(23) MeVand 5.454(4) MeV. These three levels correspond to the 5.0893(8) MeV 1+, 5.2931(14)MeV 4+ and 5.4524(4) MeV 3+ states from Ref. [16], respectively. In Table 5.1 we seean excellent agreement between our data and all previous results. We did not observe theunnatural-parity 2− level at 5.2960(4) MeV from Ref. [16]. The only unnatural-paritystates which we observed are the 5.092 MeV 1+ and 5.454 MeV 3+ with very low statis-tics, as can be seen in Fig. 5.2. In Fig. 5.3 the mirror assignments for levels below theproton-emission threshold are presented.

The only astrophysically important state in this interval is the state at 5.7110(10) MeV,which will be discussed in the next section together with theother levels above the proton-emission threshold.

5.4. Region above the proton-emission threshold (5.5042 MeV- 8.142 MeV) 65


24Mg(p,t) 22MgC

ount

s/ 1

keV

3.30

82*

4.40

20*

5.03

54*

5.71

10*

1.24

72*

g.s.

*

5.45

45.

295

14O

g.s

.

10C

g.s.

Θ=−0.3ο

Θ=8ο

Θ=17ο

ρ

Figure 5.2: 24Mg(p,t)22Mg spectra encompassing the calibration region and taken atspectrometerangles−0.3, 8 and 17. The calibration lines are marked with∗. The determined excitationenergies for22Mg are listed in the second column of Table 5.1.

5.4 Region above the proton-emission threshold (5.5042MeV- 8.142 MeV)

The excellent energy resolution of about 13 keV (FWHM for theground state) achievedin our experiment allowed us to separate close peaks and to clarify some uncertaintiesfrom previous experiments. The presently measured22Mg excitation energies between theproton-emission and alpha-emission thresholds are listedin column 3 of Table 5.2. In Fig.5.4 we show our triton spectra for this energy range taken at spectrometer angles−0.3,8, and 17.


3.3082 4+3.358 4+

5.146 2−

5.452 3+

1.2472 2+

5.641 3+5.523 (4+)5.363 2+5.331 1+

4.456 2+

1.275 2+

Ne

5.296 2−5.2938 4+

5.0354 2+

4.4020 2+

22Mg

g.s. 0+

22

5.0893 1+

g.s. 0+g.s.*

1.2472*

3.3082*

4.4020*

5.0354*(5.092(5))5.2947(23)

5.454(4)

present(p,t)(MeV)

adoptedenergy(MeV)

Figure 5.3: The possible22Mg mirror assignments for states below the proton-emissionthreshold.The 22Mg spin-parity assignments without bracket are taken from previous experiments [61]. Thefull arrows indicate the mirror assignments between22Mg and 22Ne levels for which spin-parityvalues are already known. The22Ne spin-parity values are taken from Ref. [61]. These spin-parityassignments will be used for reaction-rate calculations. Adopted levels with a dashed line have notbeen resolved in this experiment but their excitation energies have been taken from the literature.

Previous experimental results in this region are listed in columns 4 - 11 of Table 5.2.Our results agree with previously reported excitation energies. In the first column spin-parity values from previous experiments are listed as adopted in Ref. [61].

In the second column of Table 5.2 we list the spin-parity values obtained by mirrorassignment from22Ne, see Fig. 5.5. Note that the superscriptT means that the spin-parityassignments for some22Ne levels are not known, and are taken from Ref. [60].

In Fig 5.5 we show22Mg levels and the corresponding22Ne mirror states up to the


alpha-emission threshold. The problem in the spin-parity assignments is the lack of22Mgspin-parity data; the highest excitation energy for a statewith known spin and parity isthat of the 0+ state at 7.2183 MeV. For all levels above this energy mirror assignmentsare very uncertain. In the following and in later sections wewill discuss some of the levelsindividually based on the following criteria: 1) levels which differ significantly in excitationenergies from those reported in the literature; 2) levels that are of astrophysical importance;and 3) levels belonging to doublets that have not been resolved earlier.

6.0362(8) MeV (3−): Our measured excitation energy for this level is lower comparedto previous experiments, but still in agreement within errors. Spin-parity of this level re-mains uncertain. Because this state is strongly populated in the (p,t) experiments we canassume that it has natural-parity. Seweryniaket al. [16] suggest spin-parity 3−, whichcorresponds well with the 5.910 MeV, 3− state in22Ne (see Fig. 5.5).

6.2261(10) MeV (4+): Since Batemanet al. [11] measured a peak at 6.241 MeV witha relatively large width of 26±6 keV; they suggested that it consisted of a doublet. Thisconjecture was confirmed by Seweryniaket al. [16] when they resolved the doublet bymeasuring a 6+ state at 6.254 MeV. In the present (p,t) experiment we observed only alevel at the lower energy of 6.2261(10) MeV with a width of 13 keV corresponding to theenergy resolution of the present experiment. This level probably corresponds to the22Ne4+ state at 6.346 MeV (Fig. 5.5). The 6+ state at 6.254 MeV is not observed in the presentexperiment as expected because of the high momentum transfer necessary to populate thislevel strongly.

6.306(9) MeV (3+) or (1+): This level is reported in Refs. [13, 18, 19] at an energyabove 6.325 MeV. Our data are consistent with the corrected data of Batemanet al. (5th

column of Table 5.2). In our reaction this peak is weakly excited and in addition affected bythe12C and16O contaminant lines; at the spectrometer angles of−0.3 and 17 this peakcoincides with the contaminant peaks. Batemanet al. [11] reported a natural parity forthis level. The weak population is an indication of an unnatural-parity state. The possiblemirror level in22Ne is the 6.635 MeV 3+ state (Fig 5.5); it has a mirror energy shift whichis consistent with the lower lying 3+ state at 5.454 MeV. However, Fortuneet al. [63] andRuiz et al. [19] correlated this level, on basis of Thomas-Ehrmann shift calculations, to bethe mirror of the 6.855 MeV 1+ state in22Ne. Taking into consideration our assignmentsthe 22Ne 6.855 MeV state would be the first uncorrelated unnatural-parity 22Ne state inFig. 5.5.


5.95

4

6.22

6

6.76

96.

876

7.21

8

7.74

1

7.59

9

7.92

1

8.00

7

6.30

6

6.57

86.

602

7.07

97.

060

7.04

57.

027

7.33

8

7.38

9

5.71

1*

6.03

624Mg(p,t) Mg

Cou

nts/

2.5

keV

Θ=−0.3 ο

Θ=8 ο

Θ=17 ο


22

Figure 5.4: The 22Mg spectra above the proton-emission threshold. A∗ indicates a peak used forcalibration. The excitation energy in MeV for each peak is marked in the specific spectrum, where it isdetermined, obtained at either−0.3, 8 or 17. All 12C(p,t)10C and16O(p,t)14O contaminant peakshave been subtracted. The determined excitation energies for 22Mg are listed in the third column ofTable 5.2. See further Fig. 4.5 for more details.


5.296 2−

5.452 3+

5.710 2+

5.954 0+

6.037 (3−)

6.254 (6+)6.326 (3+)

6.580 (1−)6.605 (2+)

6.769 (0,1)+

6.876 (1−)

5.146 2−

5.363 2+

5.523 (4+)

5.641 3+

5.910 3−

6.120 2+

6.234 0+

6.346 4+

6.635 3+

6.691 1−

6.819 2+6.855 1+6.900 (0,1)+

7.050 1−

6.226 (4+)

7.218 0+

7.338 (2+)

7.674 (2−)

7.340 0+7.343 (3,4)+7.406 3−7.422 (5+)7.469 (3+) T7.491 1−

7.643 2+7.664 2−7.722 3−

7.923 (2+)

8.076 (4+)

8.134 2+8.162 (3+) T

8.376 (3−)

8.452 (−)8.489 2+

7.027 (3+)T7.045 (4+)7.060 (3−)

5.293 4+5.331 1+

6.310 (6+)

5.089 (1+)

7.079 (1−)

8.062 (3+) T8.005 (3−)

7.921 (2+)

7.742 (4+)

7.601 (2+)

7.384 (3−)

22Mg 22Ne

5.9538(8)

6.036(9)

6.306(9)

6.578(7)6.602(9)

6.7688(12)

6.8760(12)

7.027(9)7.045(7)7.060(7)

7.2183(10)

7.389(12)

7.5995(29)

7.9206(15)

8.0070(14)

7.338(13)

7.079(8)

6.2261(10)

5.711 *

5.454(4)

5.2947(23)

5.092(5)

5.0354 * 5.035 2+

adoptedenergy

present(p,t)

(MeV)(MeV)

7.7411(20)

Figure 5.5: The22Mg mirror assignments for states between 5 MeV up to the alpha-emission thresh-old located at 8.140 MeV. The22Mg spin-parity assignments without brackets are taken frompreviousexperiments [61]. The22Mg spin-parity values within brackets are possible mirror assignments. Thefull (dashed) arrows indicate definite (tentative) mirror assignments. The22Ne spin-parity valuesare from Ref. [61]. The spin-parity values marked by the superscript T are from Ref. [60]. Thesespin-parity assignments will be used for reaction-rate calculations. See further Fig. 5.3 for moredetails.

70

5.22M

gdata

andtheir

astrophysicalimplications

Table 5.2: Excitation energies, spins and parities of levels in the region (5.512 - 8.142 MeV) above the proton-emission threshold.

Jπ Jπ present Ref. [16] Ref. [11] Ref. [13] Ref. [52] Ref. [12] Ref. [14] Ref. [18] Ref. [19] adoptedado.a mirrorb (p,t) (p,γ) (p,t) (3He,6He) (3He,n) (16O,6He) (4He,6He) (21Na,22Mg) (21Na,22Mg) presentc

2+ 2+ 5.7110∗ 5.7110(10) 5.7106(12) 5.7139 5.699(20) 5.711 - 5.7097(5) - 5.7101(5)d

0+ 0+ 5.9538(8) - 5.9577(25) - 5.945(20) - - 5.958(5) - 5.9542(6)- 3− 6.0362(8) - 6.0413(30) 6.051(4) - 6.041(11) 6.059(9) 6.042(13) - 6.0371(8)(4+) 4+ 6.2261(10) - 6.2411(51)g 6.246(4)g - - 6.244(9)g 6.242(1)g - 6.2261(10)e

(6+) (6+) - 6.2543(3) 6.2411(51)g 6.246(4)g 6.263(20) 6.255(10) 6.244(9)g 6.242(1)g - 6.2543(3)f

- 3+ 6.306(9) - 6.3170(60) 6.329(6) - - - 6.3255(9) 6.3290(24) 6.3256(9)- 1− 6.578(7) - - - 6.573(20) - - - 6.587(10) 6.580(6)(2,3,4)+ 2+ 6.602(9) - 6.606(7) 6.616(4) - 6.606(11) 6.606(9) 6.6053(25) 6.611(11) 6.605(7)(3−) 0+T 6.7688(12) - 6.780(14) 6.771(5) 6.770(20) 6.767(20) 6.766(12) - 6.792(17) 6.7690(12)(3−) 1− 6.8760(12) - - 6.878(9) - 6.889(10) - - 6.881(?) 6.8762(12)- 3+T 7.027(9) - - - - - - - - 7.027(9)- 4+ 7.045(7) - - - - - - - - 7.045(7)- 3− 7.060(7) - - - - - - - - 7.060(7)- 1− 7.079(8) - - - - - - - - 7.079(8)0+ 0+ 7.2183(10) - - 7.206(6) 7.201(20) 7.169(11) 7.216(9) - - 7.2176(9)- 2+ 7.338(13) - - - - - - - - 7.338(13)- 3− 7.389(12) - - 7.373(9) - 7.402(13) - - - 7.384(7)- 2+ 7.5995(29) - - 7.606(11) - - 7.614(9) - - 7.6012(27)- 2− - - - - - 7.674(18) - - - 7.674(18)- 4+ 7.7411(20) - - 7.757(11) - 7.784(18) - - - 7.7420(19)- 2+ 7.9206(15) - - - - - - - 7.9206(14)- 3− 8.0070(14) - - 7.986(16) - 7.964(16) 7.938(9) - - 8.0051(13)- 3+T - - - - - 8.062(16) - - - 8.062(16)

∗ Energy from Ref. [16] used for calibration in the present work.a Adopted Jπ values by Ref. [61].b Mirror assignment, Fig. 5.5.c Weighted average.d Only data from Refs. [16, 18, 52] are included.e Only present data considered.f Included data in the weighted average are from Refs. [12, 16,52]g Unresolved doublet.T Theoretical value taken from Ref. [60].


6.60

26.57

8

24Mg(p,t) Mg22

Figure 5.6: Doublet near 6.6 MeV observed at the 8 spectrometer angle. The upper panel showsthe two-peak fit to the data. The lower panel shows the difference between data and fit (residuespectrum). The determined excitation energies for22Mg are listed in the third column of Table 5.2.See further Fig. 4.5 for more details.

6.578(7) MeV (1−), 6.602(9) MeV (2+): This doublet was resolved by Ruizet al. [19]at 6.587(10) MeV and 6.611(11) MeV, respectively. We could resolve these two levels onlyat an angle of 8 (see Fig. 5.6). At−0.3 and 17 these lines were affected by the12C and16O contaminant lines, respectively. By mirror assignments these levels were assumed tocorrespond to the 6.691 MeV 1− and the 6.819 MeV 2+ levels in22Ne, respectively. Thisis in agreement with Ref. [19].

6.7688(12) MeV (0+, 1+): This level was observed in many previous experiments.Our result is in agreement with all previous results, exceptthat of Ref. [19]. This level isproposed to have spin-parity 3− in Ref. [10]. In contrast, the mirror state in22Ne could bethe 6.900 MeV level which has (0,1)+ assignment [61]. In the21Na(p,γ)22Mg reaction-ratecalculations we will assume the natural 0+ spin-parity value.

7.027(9) MeV (3+)T, 7.045(7) MeV (4+), 7.060(7) MeV (3−), 7.079(8) MeV (1−):These four levels are for the first time observed in our high-resolution (p,t) experiment. Wehave been able to resolve these levels at spectrometer angles of 8 and 17 (Fig. 5.7). Atthe spectrometer angle of 8 we resolved all four levels, for 17 only three. These fourlevels correspond probably to the closely-spaced levels above 7.343 MeV in22Ne (see Fig.


5.5). We exclude the mirror level at 7.422 MeV with a probable5+ spin-parity, since thisstate is unlikely to be excited in our reaction due to high spin and unnatural parity.

7.2183(10) MeV 0+: The 0+ level at 7.2183 MeV is the highest lying level in22Mg forwhich adopted values of spins and parities of excited statesare known from other experi-ments; see Ref. [10]. Therefore, it is the highest energy where we can verify our spin-parityassignments with22Ne mirror states.

In this region above the proton-emission threshold and below the alpha-emission thresh-old we observed additional levels at 7.338(13) MeV and 7.9206(15) MeV. Their possiblemirror spin-parity assignment can be seen on the right panelin Fig. 5.5.


7.04

5(6)

7.06

0(6)

7.07

9(6) 24Mg(p,t) Mg

(a)

Θ=8o

22

7.02

7(7)

24Mg(p,t)(b)

Θ=17o

Mg22

7.09

2(11

)

7.04

0(13

)

7.06

7(11

)

Figure 5.7: Levels observed in the region 7.0-7.1 MeV. Upper panel: spectrum taken at a spectrom-eter angle of 8 with a 4-level fit and the residue spectrum below it. Bottom panel: spectrum taken ata spectrometer angle of 17 with a 3-level fit and the residue spectrum below it.


Table 5.3: Excitation energies, spins and parities of levels in the region (8.14 - 10.5 MeV) above theα-emission threshold in22Mg.

Jπ present Ref. [13] Ref. [12] Ref. [14] adoptedmirror (p,t) 25Mg(3He,6He)22Mg (16O,6He) (4He,6He) presenta

(2+) 8.1803(17) 8.229(20) 8.203(23) 8.197(10) 8.1812(16)(2+) 8.383(13) 8.394(21) 8.396(15) 8.380(10) 8.385(7)(3−) 8.5193(21) 8.487(36) 8.547(18)b 8.512(10) 8.5193(20)(4+)T 8.572(6) 8.598(20) - - 8.574(6)(0+)T 8.6575(17) - 8.613(20) (8.644(18))c 8.6572(17)(4+) 8.743(14) - - - 8.743(14)(1−) 8.7845(23) 8.789(20) 8.754(15) 8.771(9) 8.7832(22)(2+) 8.9331(29) - 8.925(19) 8.921(9) 8.9318(27)(1−)d 9.082(7) - 9.066(18) (9.029(20))c 9.080(7)(4+)T 9.157(4) - (9.172(23)) 9.154(10) 9.157(4)(6+)T - - (9.248(20)) - 9.248(20)(2+)T 9.315(14) - 9.329(26) (9.378(22))c 9.318(12)(3−) 9.492(13) - (9.452(21)) 9.482(11)(2+) 9.546(15) - 9.533(24) 9.542(12) 9.542(9)(6+)T - - 9.638 9.640(10) 9.640(9)(0+) (9.70(5)) - 9.712(21) - 9.709(19)(2+) 9.752(24) - - 9.746(10) 9.7516(27)(0+) 9.861(6) - 9.827(44) 9.853(11) 9.860(5)(1+) - - 9.924(28) 9.953(13) 9.948(12)(2+) 10.087(15) - 10.078(24) (10.128(20))c 10.085(13)(3+) (10.168(9)) - 10.190(29) - 10.170(8)(2+) 10.2717(17) - 10.297(25) 10.260(10) 10.2715(17)(4+)T 10.430(19) - 10.429(16) (10.389(20))c 10.429(13)

a Weighted average.b Probably unresolved doublet of the levels at 8.5193(21) MeVand 8.572(6) MeV; notincluded in the averaging of values.c The energies of these levels differ significantly from thosemeasured in the presentexperiment and Ref. [12], except for the level at 8.644 MeV which has a tentativeassignment in Ref. [14]; these levels are not included in theweighted average.d Negative parity, in this case we assume 1−.T Spin-parity value from Ref. [60].

5.5 Region above theα-emission threshold (8.142 MeV-10.5 MeV)

The 22Mg levels above theα-emission threshold are important for the18Ne(α,p)21Na re-action in X-ray bursts. Table 5.3 lists the present results together with the results fromprevious experiments Refs. [12, 13, 14]. In the last column the adopted energies for levelsare listed, which we will use for reaction-rate calculations. In Fig. 5.8 we show our tritonspectra at spectrometer angles−0.3, 8 and 17 for this energy range.

5.5. Region above the α-emission threshold (8.142 MeV-10.5 MeV) 75

24Mg(p,t) 22Mg


Cou

nts/

5 k

eV

8.18

0

8.51

9

8.57

28.

657

8.74

38.

784

8.93

3

9.08

2

9.15

7

9.75

2

9.86

1

10.2

72

9.31

5

9.49

29.

546

10.0

87

10.4

30

(10.

168)

(9.7

00)

8.38

3

g

Θ=−0.3 ο

Θ=8 ο

Θ=17 ο

Figure 5.8: The 22Mg spectra above the alpha-emission threshold. The energy for each peakis marked in the specific spectrum, where determined, obtained at either−0.3, 8 or 17. All12C(p,t)10C and16O(p,t)14O contaminant peaks have been subtracted. The determined excitationenergies for22Mg are listed in the second column of Table 5.3. See further Fig. 4.5 for more details.

22Ne is a stable nucleus. Therefore, there is more experimental information regardingits structure available than for22Mg. The number of known levels above theα-emissionthreshold in22Ne is larger than in22Mg. However, the spin-parity information is not knownfor all observed22Ne levels above theα-emission threshold. We used the calculations fromRef. [60] in an attempt to deduce the spin-parity for22Ne levels with unknown spin-parityand subsequently for those in22Mg. Therefore, the22Mg spin-parity mirror assignmentsare uncertain. In this section we will discuss some of the levels together with the tentativespin-parity assignments.

Fig. 5.9 displays tentative spin-parity assignments for levels above theα-emissionthreshold in22Mg. A larger level density in22Ne than in22Mg can be observed. We tried


to maintain a constant energy shift for levels with the same spin-parity. This can be seenin Fig. 5.9 where a relatively small energy shift for the 6+ levels at 9.248 MeV and 9.640MeV was taken similar to that of the 6+ level at 6.2543 MeV (Fig. 5.5). For the 2+ stateswe maintained an energy shift between 300 keV and 400 keV similar to the shifts of the2+ levels with known spin-parity values at lower energies (Figs. 5.3 and 5.5). In order toestimate the uncertainty of the reaction rates due to these uncertainties in level assignments,we performed calculations with different spin assignmentsfor dominant levels,i.e. levelswithin the Gamow window.

8.1803(17) MeV (2+): This is the first level above theα-emission threshold. The ob-tained excitation energy is in agreement with that obtainedin Ref. [12], but in disagreementwith Refs. [13, 14]. For this level we assigned a spin-parity2+ assuming it to be the mirrorof the22Ne 8.489 MeV state.

8.5193(21) MeV, 8.572(6) MeV, 8.6575(17) MeV, 8.743(14) MeV, 8.7845(23) MeV:Owing to our high-resolution spectra we were able to identify five levels in the range from8.5 MeV up to 8.8 MeV. The excitation energies for these five levels are deduced fromthe measurement at−0.3, and tentative spin assignments are given in Fig. 5.9. The spinassignments can be made in different ways. We assigned 0+ spin-parity to the 8.6575(17)MeV level, because of the fast decline of the cross section with increasing scattering angle(Fig. 5.10) as expected for anL=0 transition.

Only levels whisch differ significantly in excitation energy from the previous exper-iments are discussed here. The level at 10.168(9) MeV is tentatively introduced here,because of consistent analysis at the three different scattering angles. Furthermore, wedetermined the excitation energies for some of the levels inthis region with an error assmall as a few keV. This achievement is a great improvement owing to the high-resolutionspectroscopy of the GR facility. Since the excitation energies of resonance levels enterexponentially in the rate calculations, this achievement will greatly decrease the errorsoriginating from these uncertainties.

5.5. Region above the α-emission threshold (8.142 MeV-10.5 MeV) 77

8.005 (3−)

8.062 (3+)T

8.574 (4+)T

8.743 (4+)8.783 (1−)

8.932 (2+)

9.157 (4+)T

9.248 (6+)T

9.482 (3−)

9.542 (2+)

9.640 (6+)T

9.709 (0+)

9.860 (0,1,2)

9.948 (1+)

10.085 (0,1,2)

10.170 (3+)T

10.272 (2+)

8.376 (3−)

9.508 (4+) T9.541 2+

9.652 (6+) T9.725 (3−)

9.842 (2+)

10.137 2+

9.625 (2+) T

9.324 (−) T9.250 (6+) T9.229 2+9.178 (4+)9.162 (5+) T9.097 1−9.045 (2+,3−)

8.976 (4+) a

8.899 (0+) T8.855 4+

8.596 2+8.573 (5+) T8.553 (1,2+)

8.452 (−)

8.162 (3+)T8.181 (2+)

9.752 (2+)

8.489 2+8.519 (3−)

8.740 (3−)

8.385 (2+)

10.423 (3+) T10.384 (6+) T

10.469 3−10.493 2+10.551 2+10.616 (7+) T

10.706 (0+)T10.749 (4+)T

10.921 1−

10.696 (3+) T

10.857 3−

9.318 (2+)T

22Mg 22Ne

10.066 (0+)

8.657 (0+)T

8.383(13)

9.752(24)

10.087(15)

8.0070(13)

8.1803(17)

8.5193(21)8.572(6)

8.6575(17)

8.743(14)8.7845(23)

8.9331(29)

9.082(7)

9.157(4)

9.315(14)

9.492(13)9.546(15)

(9.700(50))

9.861(6)

(10.168(9))

10.2717(17)

10.430(19)

(p,t) energyadopted present

10.429 (4+)T

9.080 (1−)T

(MeV) (MeV)

10.297 (0,1,2) b10.282 (0,1,2) b10.208 1+ b

Figure 5.9: Mirror spin-parity assignments for the22Mg levels above theα-emission threshold. Theadopted22Mg energies are taken from the last column of Table 5.3. The spin assignments of the22Mg level scheme are obtained in accordance with the mirror nucleus. The energies given betweenbrackets are tentative. See further Fig. 5.3 for more details.a The spin-parity value is from Ref. [64].b The excitation energies are from Ref. [46].T Spin-parity value is from Ref. [60].


8.51

9

8.78

4

8.74

3

Θ=−0.3o

8.57

2

8.65

7

oΘ=8

Figure 5.10: 24Mg(p,t)22Mg spectra taken at spectrometer angles−0.3 and 8 with fits to the fiveresolved levels between 8.5 MeV and 8.8 MeV. The determined excitation energies for22Mg arelisted in the second column of Table 5.3. See further Fig. 4.5for more details.

5.6. Region above 10.5 MeV 79

Table 5.4: Excitation energies, spins and parities of levels in the region above 10.5 MeV.

Jπ Present Ref. [12] Ref. [20] Ref. [21] Ref. [14] adoptedmirror (p,t) (3He,6He) (18Ne,p) (18Ne,p) (4He,6He) presenta

(3+)T - 10.570(25) (10.580(50)) 10.55(14) - 10.572(23)(3−) 10.667(19) 10.660(28) - 10.66(14) 10.627(20) 10.651(13)(2+) 10.768(21) 10.750(31) - - 10.776(20) 10.768(13)(4+)T 10.881(15) 10.844(38) (10.820(60)) 10.86(14) - 10.873(14)(8+) - - 10.910(50) 10.92(14) (10.915(20)) 10.914(19)b

(0+) 10.999(15) 10.980(31) 10.990(50) 11.01(14) (11.015(20)) 11.001(11)b

(6,7) - - (11.050(50)) - - 11.050(50)(7−) - 11.135(40) 11.130(50) - (11.118(20)) 11.122(17)b

(6+)T - - - - (11.231(20)) 11.231(20)b

(4+)T 11.317(27) - - - 11.313(20) 11.315(16)(2+) 11.499(17) - - - - 11.499(17)(1−) 11.603(16) - - - 11.581(20) 11.595(12)(0+) 11.76(3) - - - (11.742(20)) 11.747(17)b

(0+) 11.937(17) - - - 11.881(20) 11.914(13)(1−) - - - - (12.003(20)) 12.003(20)b

(3−) 12.220(30) - - - (12.169(20)) 12.185(17)b

(2+) 12.474(26) - - - - 12.474(26)(3−) 12.665(17) - - - - 12.665(17)(0+) (13.010(50)) - - - - (13.010(50))

a Weighted average.b Here, we included the tentatively determined levels from Ref. [14].T Spin-parity from Ref. [60].Excitation energies within brackets indicate tentativelyidentified levels.

5.6 Region above 10.5 MeV

The22Mg levels in the region above 10.5 MeV become important for X-ray bursts at peaktemperatures around 2.5 T9. Previous studies [12, 14, 20, 21] succeeded in identifyingsev-eral levels in this region. The resolution achieved in previous experiments was, however,insufficient to resolve most states. For this energy region,we have performed measure-ments at scattering angles of 8 and 17. Our results together with the data from theseprevious experiments are listed in Table 5.4.

From our data (listed in the second column of Table 5.4) it canbe seen that the accuracyfor the determination of the peak position is worse comparedto those for levels at lower ex-citation energy as discussed in the previous sections. The increase of the excitation-energyerror is due to the error in the scattering-angle determination (0.15); this error contributesmore than 5 keV or 10 keV for the scattering angles of 8 or 17, respectively. However,due to the larger statistics collected at the 17 scattering angle than at the 8 scattering an-gle, the total error is comparable for both scattering angles. A level was adopted if identified



24Mg(p,t) 22Mg

10.6

6710

.768

10.8

8110

.999

11.3

17

11.4

9911

.603

11.7

58

11.9

37

12.2

21

13.0

10

Cou

nts/

10

keV

l

12.6

65

12.4

74

Figure 5.11: The24Mg(p,t)22Mg spectrum above 10.5 MeV, measured at a scattering angle of17.The determined excitation energies for22Mg are listed in the second column of Table 5.4. See furtherFig. 4.5 for more details.

at both scattering angles. Because of the larger collected statistics we use22Mg excitationenergies determined from the 17 spectrum. The results are listed in the second column ofTable 5.4. Fig. 5.11 presents a spectrum taken at 17 where all impurity lines have beensubtracted.

In the last column of Table 5.4 we list the averaged energies from all experiments,including the tentatively identified peaks from Ref. [14]. We confirmed the existence ofthe level at 11.742(20) MeV which was tentatively identifiedin Ref. [14].

In the region above 10.5 MeV there is no previous spin-parityinformation for22Mglevels. Furthermore, there is very little experimental information from the22Ne mirrornucleus in this excitation-energy region; additional Jπ information were taken from thecalculations of Brown [60]. The level structure of22Mg is given in Fig. 5.12; it presents

5.6. Region above 10.5 MeV 81

one possible choice that was used as a starting point to studythe influence of the paritiesand spins on the calculated reaction rates.

The excitation energies and corresponding Jπ values of levels in22Ne are taken fromRef. [61] and they will be the leading source of the data for correlation with the22Mgmirror nucleus. Additional information is taken from Ref. [62]. These additional dataare marked with# in Fig. 5.12. Some of the levels from Ref. [62] are introducedin the22Ne level scheme, see Fig. 5.12 (dotted lines), and some are associated with the adopteddata from Ref. [61]. By comparing the data from Refs. [61, 62]it can be seen that thereis a discrepancy in the energy and the spin-parity values forsome of the states. If noexperimental information on spin-parity was given in Ref. [61], we used data from Ref.[62]. In the remaining cases we used the calculation given byBrown [60].


10.429 (4+)T

10.572 (3+)T

10.873 (0+)T10.914 (8+)

11.001 (0+)T

11.122 7

11.315 (4+)T

11.747 (0+)

11.914 (0+)

12.003 (1−)

12.185 (3−)

10.282 (0,1,2)10.297 (0,1,2)

10.384 (6+) T10.423 (3+)T10.469 3−10.493 2+10.551 2+10.616 (7+)T10.696 (3+)T10.706 (0+)T10.749 (4+)T

10.857 3−

10.921 1−

11.032 (8+)11.064 2+11.130 (6,7)

11.194 (0+)T

11.465 711.433 (4,0,2+)T

11.271 (0+)T

12.665 (3−)

11.533 (6+)T11.577 (4+)T

12.000 (4+) T

12.450 (0+,1−)

12.862 (3−)

12.910 (1+) T

13.078 (4+) T

13.274 (0+) T

13.384 (4+) T

10.272 (2+)

10.651 (3−)

10.768 (2+)

12.474 (2+)

11.050 (6,7)

11.595 (1−)

11.231 (6+)T

11.656 (1+) T

12.820 (1−)#12.800 (2+)#12.700 (3−)#12.643 (2+)#

12.390 3− (2+)#

12.280 (1−)#

11.772 3− (1−)#

11.708 (1−) (2+)#

12.218 (0+)#

12.570 (1−)#

11.896 (1−)#

12.071 (0+)#

22Mg 22Ne

11.499 (2+)

13.010 (0+)T(13.010(50))

12.665(17)

12.220(30)

11.760(30)

11.317(27)

10.999(15)

10.881(15)

10.768(21)

10.667(19)

10.2717(17)

11.499(17)

11.603(16)

11.937(17)

12.474(26)

adoptedenergy(MeV)

present(p,t)(MeV)

10.430(19)

Figure 5.12: Spin-parity assignments for the22Mg levels above 10.5 MeV. The adopted excitationenergies plotted for22Mg are those listed in the last column of Table 5.4. The data for 22Ne are fromRefs. [61, 62]. See further Fig. 5.3 for more details. We use the following notation:T spin-parity from Ref. [60].# data taken from Ref. [62].

5.7. Astrophysical implications for the 18Ne(α,p)21Na reaction 83

5.7 Astrophysical implications for the18Ne(α,p)21Na reac-tion

Several experiments [12, 13, 14, 20, 21] were performed to determine properties such asthe excitation energy, the spin and parity values, and the resonance strengths of22Mg statesabove the alpha-emission threshold (8.142 MeV). Together with our data, the excitationenergies of 42 levels have been determined up to now. However, we are still lacking in-formation regarding the spin and the resonance strength formany of these levels. In thisparagraph we will calculate the reaction rates for the18Ne(α,p)21Na reaction by using22Mg adopted excitation energies and discuss possible errors.

Each of the presently measured resonances, has a resonance width which is less than10% of its resonance energy. The only exception is the level presently measured at Ex=9.082MeV, where the measured width is 12.5% of the resonance energy:Γ/Eres=12.5%. Be-cause of this we will use the theoretical description outlined in Ref. [4] in the case ofnarrow resonances. According to this description, the total reaction rate can be expressedas the sum of those of individual resonances, as it is explained in Section 2.3.

It can be seen from Eq. 2.30 that the largest uncertainties inthe reaction-rate calcula-tions are introduced by uncertainties in the resonance energy, because of the exponentialdependence. However, with the present accuracy with which resonance energies are deter-mined, the larger error in the21Na(α,p)21Ne rate calculations is introduced by the unknownresonance strengths. In Refs. [20, 21] resonance strengthsfor seven and eight levels, re-spectively, have been measured and they differ by up to an order of magnitude. However,in both of these experiments spin information for the observed levels was not obtained.

The data determined by Groombridgeet al. [21] are given in Table 5.5. These authorswere able to resolve eight22Mg levels which are in good agreement with the adoptedvalues, the only exception being the first level which may be the unresolved doublet of10.087 MeV and 10.168 MeV. In contrast, Bradfield-Smithet al. [20] identified only twolevels. Groombridgeet al. [21] admitted that the intrinsic resonance widths (Γ) are muchnarrower than observed in their experiment because of the limited experimental resolution.Because of the large difference in the observed resonance strengths between Refs. [21] and[20] we will use the data for the last seven levels given in Ref. [21] as an upper limit.

Sources for the Sα values are calculations by Gorreset al. [64] that are listed in Table5.6. Data regardingα-cluster structure of22Ne from Ref. [62] are listed in Table 5.7. In theprevious section we mentioned a discrepancy between the data published in Refs. [62] and[61], which are both listed in Table 5.7. In cases where we didnot have any experimentalinformation, we used the values of Sα calculated by Hess [65].

The lack of data for theα-spectroscopic factors in22Mg is obvious from Tables 5.5- 5.7. We decided to use the Sα values for22Ne from Tables 5.6 and 5.7 for the mirrorlevels in22Mg listed in the same tables. In the following, these Sα values will be denoted


Table 5.5: The resonance strengths for the18Ne(α,p)21Na reaction taken from Ref. [21].

Ex(22Mg) Adopted(MeV) ωγ (eV) Ex (MeV)10.12(14) 1400 -10.31(14) 10300 10.27210.42(14) 7300 10.42910.55(14) 18800 10.57110.66(14) 18200 10.655- - 10.76810.86(14) 45200 10.87610.92(14) 34000 10.91411.01(14) 8100 11.001

Table 5.6: Resonance parameters for the18Ne(α,p)21Na reaction taken from Ref. [64].

Ex(22Ne) Jπ Sα Ex(22Mg) ωγ(MeV)a 22Ne (MeV) (eV)8.489 2+ 0.003 8.181 5.75E-678.596 2+ 0.225 8.385 8.94E-188.740 3− 0.012 8.519 1.43E-138.976 4+ 0.060 8.743 1.40E-89.097 3− b 0.050 8.783 4.76E-79.725 3− 0.080 9.484 6.78E-19.842 2+ 0.090 9.543 5.76E+010.066 0+ c 0.400 9.707 1.36E+2

a These are the possible22Ne mirror levels.b This level is listed as 1− in Ref. [61], and the Sα will not be used in the presentcalculation for the reaction rate.c This level has negative parity in Ref. [61], and the Sα will not be used in the presentcalculation for the reaction rate.

asacquired spectroscopic factors. For the remaining levels we will adopt a constant valuefor each specific spin state. These fixed values are chosen on basis of the Sα values listedin Tables 5.5, 5.6 and 5.7 and the calculations taken from Ref. [65], see also Fig. 5.13.We adopted a value around the smaller Sα value for a particular spin. For example, for22Ne 2+ states at 9.842 MeV, 11.700 MeV, 12.610 MeV, 12.800 MeV and 13.030 MeVexcitation energy Sα values are 0.090, 0.026, 0.017, 0.016 and 0.045, respectively. Weadopted Sα=0.020 to be a reasonable approximation for the constant spectroscopic factors,for all levels with adopted spin-parity of 2+. Fig. 5.13 shows the known Sα values for thenatural-parity states up to 6+ and the adopted constant Sα values per spin value. Theseconstant Sα values are listed in Table 5.8, and will be denoted in the following as theconstant spectroscopic factors.


Table 5.7: Resonance parameters for the18O(α,γ)22Ne reaction from Ref. [62].

Ex(22Ne)a Ex(22Ne)b Eres(c.m.) Jπ Γα Sα Ex(22Mg) Eres ωγ(MeV) (MeV) (MeV) (eV) adopted (MeV) (MeV) (eV)11.700 11.708 2.040 2+ 500 0.026 11.499 3.357 2.04E+411.760 11.772 2.104 3− a 3080 0.541 - - -11.880 11.896 2.228 1− 3100 0.031 11.597 3.455 3.52E+412.020 12.000 2.332 0+ 44880 0.207 11.749 3.607 1.35E+512.250 12.218 2.550 0+ 76000 0.208 11.922 3.780 1.68E+512.280 12.280 2.612 1− 5100 0.018 12.003 3.861 3.69E+412.390 12.390 2.722 3− a 5940 0.120 12.192 4.050 1.76E+512.570 - 2.902 (1−) 36750 0.074 - - -12.610 12.643 2.975 (2+) 4960 0.017 12.474 4.523 6.01E+412.700 - 3.032 3− 900 0.008 - - -12.800 - 3.132 2+ 6000 0.016 - - -12.820 - 3.152 1− 90100 0.124 - - -12.890 12.910 3.242 3− 3510 0.021 - - -12.990 - 3.322 0+ 32000 0.028 - - -13.030 13.078 3.410 2+ 26100 0.045 - - -13.190 - 3.522 3− 45820 0.165 - - -13.490 - 3.822 4+ 2900 0.022 - - -13.690 - 4.022 (5−) 2000 0.050 - - -

a Data taken from Ref. [62].b Data taken from Ref. [61].All Jπ values are taken from [62], except values marked witha which are taken from Ref.[61].Sα factors are calculated on basis of the resonance energies,J andΓα listed in the third,fourth and fifth columns, respectively, using Eq. 2.29.The22Mg resonance strengthsωγ are calculated on basis of resonance energies,J andSα

tabulated in the eighth, fourth and sixth columns, respectively, and,Jpro=0,Jtar=0, byusing Eq. 2.28.

Table 5.8: Theconstant Sα values assumed for the levels without any experimental information.

J Sα

0 0.0301 0.0252 0.0203 0.0104 0.0605 0.0506 0.025

By using the already known Sα values listed in Tables 5.6 and 5.7 (acquired Sα) andconstant values for a particular spin of the states with unknown Sα values, we attempt tomaintain the simple procedure followed in this work. With the constant α-spectroscopic


8

14

14

14

14

14

8

8

8

8

8

8

0.02

6

0.01

7

0.22

5

0.00

3

0.09

0

14

14

0.40

0

0.20

8

0.20

7

0.02

8

0.03

1

0.01

8

0.01

2

0.05

0

0.12

0

0.00

8

0.06

0

0.05

0

0.09

1

0.00

2

0.02

2

22Mg excitation energy (MeV)

0+

1−

2+

3−

4+

5−

6+

Jπ

0.030

0.025

0.020

0.060

0.050

0.025

0.010

Acquired S Constant Sαα

Figure 5.13: Theacquired Sα values for the spin values from 0 up to 6 for the entire astrophysicallyimportant region of excitation energies in22Mg. Sα values are taken from Tables 5.6 and 5.7; all Sα

values for the spin value 6 are taken from Ref. [65].Constant Sα factors are chosen on basis of theacquired Sα values and they are indicated on the right side of the figure.

factors for the levels with the same spin and parity it will beeasy to compare and improvethe calculations once accurate experimental data will become available. With the choice oflow Sα values we attempt to be on the low limit of the reaction rates.

The mirror spin assignments given in Tables 5.3 and 5.4 and inFigs. 5.9 and 5.12 areused in our calculations of the reaction rates because thesespin assignments allow us touse the22Ne known Sα values. However, these spin assignments are arbitrary and we willcheck the influence of the spin assignments by comparing our mirror assignments with arandomly generated spin distribution. Three possible combinations are given in Table 5.9,with the last two spin assignments being randomly generatedaccording to the followingprocedure. The upper limit for a randomly generated spin was6. This number was chosenbecause the ratio of the penetrabilities for states having spin 0 and 6 can go up to 10000,which is already a noticeable difference. Furthermore, equal probability for all possiblespins is assumed. The second restriction was to preserve theacquired Sα values in bothrandom cases (Table 5.9). Thus we assume that the corresponding levels have a properlydetermined spin and Sα values.

Calculated reaction rates for the astrophysically interesting interval of temperature are


Table 5.9: The spin values and resonance strengths for the18Ne(α,p)21Na reaction.

mirror a RND1 b RND2 b

Eres (MeV) J ωγ (eV) J ωγ (eV) J ωγ (eV)

0.0391 2 9.14E-67c 2 9.14E-67c 2 9.14E-67c

0.2431 2 9.35E-18c 2 9.35E-18c 2 9.35E-18c

0.3773 3 1.46E-13c 3 1.46E-13c 3 1.46E-13c

0.4317 4 3.26E-12 1 5.93E-11 6 2.10E-160.5152 0 1.36E-08 3 7.84E-11 6 2.42E-140.6010 4 1.40E-08c 4 1.40E-08c 4 1.40E-8c

0.6415 1 2.75E-06 6 5.30E-12 3 1.37E-80.7901 2 7.51E-05 6 5.58E-10 0 1.64E-40.9383 1 5.26E-03 4 1.75E-05 6 1.91E-81.0149 4 6.70E-04 0 1.60E-02 5 3.98E-61.1060 6 5.75E-06 1 2.90E-02 6 4.43E-71.1756 2 9.04E-02 0 1.76E-01 6 1.35E-61.3419 3 6.66E-01c 3 6.66E-01c 3 6.66E-1c

1.4012 2 5.72E+00c 2 5.72E+00c 2 5.72E+0c

1.4976 6 1.07E-03 2 6.52E-01 4 4.33E-21.5651 0 1.43E+01 3 1.15E-01 5 5.75E-31.6097 2 8.65E+00 6 2.60E-04 6 2.60E-41.7176 0 4.36E+02c 0 4.36E+02c 5 4.36E+2c

1.8059 1 8.48E+01 0 5.94E+01 6 1.52E-31.9435 2 8.99E+01 4 1.60E+00 1 6.55E+12.0263 3 2.25E+01 3 3.22E+00 3 3.22E+02.1295 2 1.03E+04e 4 1.03E+04e 6 1.03E+4e

2.2874 4 7.30E+03e 6 7.30E+03e 4 7.30E+3e

2.4295 3 1.88E+04e 6 1.88E+04e 3 1.88E+4e

2.5133 3 1.82E+04e 5 1.82E+04e 2 1.82E+4e

2.6263 2 2.11E+03 3 5.52E+01 3 5.52E+12.7338 0 4.52E+04e 2 4.52E+04e 1 4.52E+4e

2.7724 6 3.40E+04e 6 3.40E+04e 1 3.40E+4e

2.8587 0 8.10E+03e 0 8.10E+03e 2 8.10E+3e

2.9080 6 1.24E+01 1 2.98E+03 0 5.37E+32.9804 6 1.66E+01 1 3.58E+03 6 1.28E+03.0890 6 2.55E+01 1 4.60E+03 2 1.69E+33.1732 4 3.66E+03 4 4.07E+02 2 2.08E+33.3572 2 2.04E+04d 2 2.04E+04d 2 2.04E+4d

3.4553 1 3.51E+04d 1 3.51E+04d 1 3.51E+4d

3.6072 0 1.34E+05d 0 1.34E+05d 0 1.34E+5d

3.7798 0 1.67E+05d 0 1.67E+05d 0 1.67E+5d

3.8610 1 3.69E+04d 1 3.69E+04d 1 3.69E+4d

4.0495 3 1.73E+05d 3 1.73E+05d 3 1.73E+5d

4.3322 2 6.01E+04d 2 6.01E+04d 2 6.01E+4d

4.5226 3 2.88E+04 2 1.72E+04 6 1.21E+24.8650 0 5.88E+04 6 2.44E+02 3 6.08E+3

a Spin and resonance strength for the mirror assignments given in Figs. 5.9 and 5.12.b Spin and resonance strength for the randomly generated spins of states.c Sα factors taken from Table 5.6, andd Sα factors taken from Table 5.7.e Resonance strengths are measured in Ref. [21].c, d ande denote levels withacquired Sα.


presented in Fig. 5.14. Previous calculations given in Refs. [12, 20, 21, 64] are alsoplotted for comparison. Our calculated18Ne(α,p)21Na reaction rates are at least five timelarger than any of the previous calculations, for a stellar temperature larger than 0.3 T9

(T9 ≡109 K). In Fig. 5.14, it can be noticed that the reaction-rate calculation done byGroombridgeet al. [21] has a similar shape to the present calculation for temperaturesabove 1 T9. The reason for the shape similarity between our work and thecalculationsfrom Refs. [12, 20, 21, 64] are theacquired Sα values taken from those references. Thefirst-excited level above theα-emission threshold does not contribute significantly to thecalculated reaction rates because it is below the Gamow window for the 18Ne(α,p)21Nareaction (see Section 2.5) at the predicted X-ray burst temperatures.

Fig. 5.15 displays the contribution of levels with theacquired Sα values taken fromdifferent Refs. [21, 62, 64]. The low contributions of the levels with constant Sα can beobserved from Fig. 5.15; this effect is due to the much lower Sα values of these levels.

The largest difference between18Ne(α,p)21Na reaction rates, calculated for the ran-domly produced spin values compared to the assigned mirror spin values was found to beabout 70%. This is, for example, the case if for one level in the first seta spin 0 valueand in the second set a spin 6 value was used. In that case, the difference in the penetra-bilities through the Coulomb and orbital-momentum barriers can differ by five orders ofmagnitude. Because in the present experiment we did not clearly observe any 6+ state, itis unlikely that we are making a huge error in the22Mg spin-parity assignments (i.e. spin0 instead of spin 6 and vice versa). Therefore, we are limiting the error originating fromthe spin-parity assignments to less than an order of magnitude. In Fig. 5.16, it can be seenthat there is almost no difference between these various choices made for the spin valuesfor a stellar temperature above 1 T9. The reason for this behavior is coming from the fixedresonance strengths for levels marked by the superscriptsa, b andc in Table 5.9 (levels withtheacquired Sα).

With the assumption that our mirror spin assignments and corresponding Sα values arecorrect we calculated the error for the18Ne(α,p)21Na reaction rate caused by the uncer-tainty of the resonance energy (Table 5.10). The large relative error at low temperatures(0.1-0.2 T9) is caused by the energy error for the 8.385(7) MeV level which dominates atthese temperatures. With increasing temperature the errororiginating from the resonance-energy uncertainties drops to 2%, and it is significantly smaller than uncertainties whichoriginate from the spin assignments and Sα values.

We calculated the18Ne(α,p)21Na reaction rate with the presently available data set.The values obtained can change due to several reasons:

1. There may be missing states in22Mg which can significantly contribute to the18Ne(α,p)21Na reaction. We already concluded before that the observed level densityof 22Ne is larger than that of22Mg in the region above theα-emission threshold.

2. The spin assignment can be incorrect especially at the higher excitation energies. We


0.1 1 10T

9 (K)

10-25

10-20

10-15

10-10

10-5

100

105

Rea

ctio

n ra

tes

(cm

3 m

ol-1

s-1

)

presentChen et al. [12]Bradfield et al. [20]Groombridge et al. [21]Görres et al. [64]

18Ne(α,p)

21Na

Figure 5.14: The 18Ne(α,p)21Na reaction rates as a function of temperature. The different curvesindicate reaction rates calculated in Refs. [12, 20, 21, 64]as well as the one calculated in the presentwork with theacquired Sα from those references.

0.1 1 10T

9 (K)

10-15

10-10

10-5

100

105

Rea

ctio

n ra

tes

(cm

3 m

ol-1

s-1

)

presentGoldberg et al. [61]Görres et al. [64]Groombridge et al. [21]constant Sα

18Ne(α,p)

21Na

Figure 5.15: Contributions from levels with the acquired Sα from Refs. [21, 62, 64] and the constantSα to present18Ne(α,p)21N reaction rates.


Table 5.10: The relative errors of the18Ne(α,p)21Na reaction rate originating from the uncertaintyin resonance energy.

Temperature NA < σv > relative errorT9 (K) (cm3 mol−3 s−1)

0.1 4.14E-24 0.730.2 4.92E-16 0.100.3 1.30E-11 0.080.4 4.52E-09 0.080.5 2.90E-07 0.100.6 7.63E-06 0.130.8 1.57E-03 0.171.0 7.98E-02 0.152.0 1.06E+03 0.063.0 4.76E+04 0.044.0 3.43E+05 0.035.0 1.15E+06 0.036.0 2.60E+06 0.028.0 7.23E+06 0.0210.0 1.31E+07 0.02

have already shown that wrong spin assignments can change the reaction rates by upto one order of magnitude.

3. There is lack of experimental information on theα-spectroscopic factors for the22Mg states.

4. The accuracy of the determination of the excitation energy can be improved.

Our experiment significantly improved the precision of the measured22Mg excitationenergies for levels above the proton-emission threshold. Consequently, the reaction-rateerrors originating from the error in the spin assignments can reach an order of magnitudeand this is larger than the error caused by the excitation-energy precision (less than 20% forthe stellar temperatures above 0.1 T9). Therefore, more experimental efforts are necessaryto obtain the spin and Sα values for the22Mg levels.

5.8. Astrophysical implications for the 21Na(p,γ)22Mg reaction 91

0.1 1 10T

9 (K)

10-25

10-20

10-15

10-10

10-5

100

105

Rea

ctio

n ra

tes

(cm

3 m

ol-1

s-1

)

mirrorRND1RND2

18Ne(α,p)

21Na

Figure 5.16: The 18Ne(α,p)21Na reaction rate as a function of temperature for the three differentsets of spin assignments given in Table 5.9.

5.8 Astrophysical implications for the 21Na(p,γ)22Mg re-

action

The 21Na(p,γ)22Mg reaction has been studied in recent experiments; see Refs. [16, 19].These experiments significantly improved our knowledge of22Mg. We already emphasized(Section 4.3) the error introduced by using an excitation energy of 5.7139 MeV for thefirst level above the proton-emission threshold in the calibration procedure. By avoidingthis calibration problem and owing to the excellent resolution of the present experiment,the excitation energies of the levels in22Mg above the proton-emission threshold weredetermined with better precision; see Section 5.4.

By using the present data together with the data of the resonance strengths given inRefs. [17, 18, 19] we performed reaction-rate calculationsfor the21Na(p,γ)22Mg reactionin the range of stellar temperatures between 0.01 T9 and 5 T9. In the case of ONe novaewe are interested in temperatures up to 0.4 T9, corresponding to excitation energies up to5.95 MeV in22Mg. Similarly, in the case of X-ray bursts we are interested in temperaturesup to 2.5 T9 corresponding to excitation energies up to 7.2 MeV.

In Table 5.11 we list levels in22Mg above the proton-emission threshold together withtheir resonance strengths. The resonance strengths for thefirst nine levels above the proton-


emission threshold (except for Ex=6.226 MeV level) were previously measured using aradioactive21Na beam at the TRIUMF facility [17, 18, 19]. These eight resonance strengthsare marked by the superscriptsa, b and c, respectively. In Ref. [66], it is shown thatΓp ≪ Γγ only holds for the 5.710 MeV level. Already for the 5.954 MeV levelΓp ≫ Γγ

and this is also the case for all higher levels. Therefore, for these levels the resonancestrength

ωγ =2J + 1

(2Jpro + 1)(2Jtar + 1)

ΓpΓγ

Γtot(5.2)

can be expressed in terms ofΓγ only, sinceΓp ≈ Γtot:

ωγ =2J + 1

(2Jpro + 1)(2Jtar + 1)Γγ . (5.3)

In the mirror nucleus22Ne the proton, neutron andα-emission thresholds are at 15.266MeV, 10.364 MeV and 9.699 MeV, respectively. Therefore, we used the experimentallydetermined half-life data for decay viaγ-emission only [61] to calculateΓγ of 22Ne levels.Furthermore, we corrected these values for the difference in γ-ray transition energies of themirror transitions; those levels are marked by the superscript d in Table 5.11. For the22Nelevels where no multipolarity information exists for the emittedγ-rays we took the smallermultipolarity values which satisfied the selection rules for the transition. In cases where anemittedγ-ray has a mixed multipolarity, but the multipolarity mixing ratio was not given inRef. [61] and no mixing multipolarity was assumed, we took the multipolarity which gavethe smallerΓγ after correction. This is done in order to ensure that our calculations havea smaller contribution from the unknown parameters, similar to the choice of the constantSα factors for the18Ne(α,p)21Na reaction-rate calculations. For the levels indicated bythe superscripte in Table 5.11 no half-life information exists for the mirrorlevels in22Ne.Therefore, we assumed the lower value ofΓγ=0.01 eV. The calculated resonance strengthsare also listed in Table 5.11.

D’Auria et al. [18] calculated the reaction rates using the method of narrow resonances,except for the resonances at 0.8214 and 1.1012 MeV, where broad resonances were used.They did not find a significant difference between the above-mentioned method and themethod of narrow resonances over the temperature region associated with novae. Ruizetal. [19] investigated radiative capture through the first-excited state of21Na at 0.332 MeV;this state will be populated at astrophysical temperatures. Their result shows a negligi-ble difference between the total reaction rate using as seedonly 21Na in its ground statecompared to a calculation with an additional thermal population of the first-excited state,followed by subsequent capture. For these reasons and because the total widthΓ is smallerthan 10% of the resonance energy for all resonances in the excitation-energy region ofinterest we used the narrow-resonance treatment, described in Ref. [4].


Table 5.11: The21Na(p,γ)22Mg reaction: resonance energies, spin-parity assignmentsand resonancestrengths.

Ex (22Mg) Eres Jπ ωγ(MeV) (MeV) mirror (eV)5.7101 0.2059 2+ f 1.03E-3a

5.9542 0.4500 0+ f 8.60E-4b

6.0371 0.5329 (3−) 1.15E-2b

6.2261 0.7219 (4+) 5.23E-2d

6.2543 0.7501 (6+) f 2.19E-1b

6.3256 0.8214 (3+) 5.56E-1b

6.5807 1.0765 (1−) 3.68E-1c

6.6054 1.1012 (2+) 8.09E-2c

6.7690 1.2648 (0+) 4.84E-3c

6.8762 1.3720 (1−) 2.29E-3d

7.0268 1.5226 (3+) 2.71E-3d

7.0447 1.5405 (4+) 1.40E-2d

7.0604 1.5562 (3−) 9.01E-3d

7.0790 1.5748 (1−) 3.75E-3e

7.2176 1.7134 0+ f 2.90E-2d

7.3382 1.8340 (2+) 7.08E-1d

7.3842 1.8800 (3−) 8.75E-3e

7.6012 2.0970 (2+) 6.25E-3e

7.6740 2.1698 (2−) 3.26E+0d

7.7420 2.2378 (4+) 1.13E-2e

7.9206 2.4164 (2+) 6.25E-3e

8.0051 2.5009 (3−) 8.75E-3e

8.0620 2.5578 (3+) 8.75E-3e

8.1812 2.6770 (2+) 6.25E-3e

a Resonance strength taken from Ref. [17].b Resonance strength taken from Ref. [18].c Resonance strength taken from Ref. [19].d Resonance strength calculated from the half-life data for22Ne mirror levels.e A 0.01 eV constant partial widthΓγ is assumed.f Jπ values taken from Ref. [61].Jπ values in brackets are obtained by mirror assignments givenin Fig. 5.5.

The direct capture (DC) contribution is calculated using the prescription for non-resonancereaction rate given in Ref. [67]. The astrophysical factor Sis taken from Ref. [11]:

S(E) = 7.9 × 10−3 − 3.4 × 10−3E + 1.8 × 10−3E2 [MeV· b] (5.4)

The total21Na(p,γ)22Mg reaction rate combining the contributions of DC processes andresonance-capture processes is shown in Fig. 5.17. At the stellar temperatures below 0.06T9 the21Na(p,γ)22Mg reaction proceeds mainly through the DC processes, whileat higher


0.01 0.1 1T

9 (K)

10-30

10-25

10-20

10-15

10-10

10-5

100

Rea

ctio

n ra

tes

(cm

3 m

ol-1

s-1

)

resonantdirectsum

21Na(p,γ)22

Mg

Figure 5.17: The total21Na(p,γ)22Mg reaction rate.

temperatures resonance-capture processes dominate. The contributions from the two mostdominant resonances are shown in Fig. 5.18.

The 0.2059 MeV resonance contribution is dominant in the entire interval of ONe novatemperatures (0.05-0.4 T9). This conclusion is in agreement with Refs. [11, 15, 18, 19].Furthermore, the same resonance dominates the production of 22Mg in X-ray bursts up toa temperature of 1.1 T9. Beyond 1.1 T9 the main contribution comes from the resonanceat a resonance energy of 0.8214 MeV. This is mainly due to the larger resonance width ascompared to near-lying other resonances. D’Auriaet al. [18] concluded that the resonanceat 0.738 MeV (Ex=6.246 MeV unresolved doublet) yields the dominant contribution be-yond a temperature of 1.1 T9. They state that the resonance at Eres=0.8214 MeV takesover at higher temperatures. The source of uncertainties inthe present calculations can bethe resonance strengths taken for the levels at Eres=0.7219 MeV and Eres=0.7500 MeV,which are calculated from the half-lives in the mirror nucleus 22Ne and from Ref. [18],respectively. This is due to the fact that D’Auriaet al. [18] did not resolve the doubletat excitation energies of 6.2261 MeV and 6.2543 MeV; insteadthey measured one level at6.246 MeV. This can be indication that D’Auriaet al. [18] measured the resonance strengthwith the contributions from both levels. However, we will use this resonance strength forthe level at Eres=0.7500 MeV. Furthermore, the level at a resonance energy of0.8214 MeVstrongly dominates above a stellar temperature of 1.1 T9.


0.1 1T

9 (K)

10-10

10-8

10-6

10-4

10-2

100

102

104

Rea

ctio

n ra

tes

(cm

3 m

ol-1

s-1

)

0.206 MeV0.821 MeVtotal reaction rate

21Na(p,γ)22

Mg

Figure 5.18: The total21Na(p,γ)22Mg reaction rate with the contributions from the two most domi-nant resonances at energies of 0.206 MeV and 0.821 MeV. See further Fig. 4.5 for more details. Thetotal reaction rate (dotted curve) is higher than the summedcontribution of the these two resonancesbeyond 1 T9.

0.1 1T

9 (K)

10-15

10-12

10-9

10-6

10-3

100

103

Rea

ctio

n ra

tes

(cm

3 m

ol-1

s-1

)

presentD’Auria et al. [18]Bateman et al. [11]Davids et al. [15]

21Na(p,γ)22

Mg

Figure 5.19: The21Na(p,γ)22Mg resonance reaction rates as a function of stellar temperature. Thepresent results are plotted in comparison with those calculated in Refs. [11, 15, 18].


Fig. 5.19 shows our calculated21Na(p,γ)22Mg resonance reaction rate in comparisonwith those calculated in Refs. [11, 15, 18]. All calculations are shown for the stellartemperatures above 0.06 T9, because DC dominates at the lower stellar temperatures. Ourpresent calculations yield almost identical results as obtained by D’Auriaet al. [18]. Thisis a result of assuming the same resonance strengths for five resonances. We did not takeinto account the contribution from the level at an excitation energy of 5.837 MeV observedonly by Ref. [68]. However, it was already shown [18, 19] thatthis resonance has a smallcontribution to the total reaction rate due to the dominanceof the first-excited level abovethe proton-emission threshold in22Mg.

The huge discrepancy between the present results and those reported by Batemanet al.[11] has to be attributed to the overestimated resonance strength for the resonance at 0.450MeV in Ref. [11] (larger by three orders of magnitude). The reaction-rate calculationsgiven by Davidset al. [15] were performed by using only two resonances, in contrast to ourwork where we included all the resonances listed in Table 5.11. Therefore, a discrepancywith the present results can be observed above 0.8 T9.

For ONe novae, the calculated21Na(p,γ)22Mg reaction rate is consistent with the cal-culations done in Refs. [15, 18]. For temperatures higher than 1.1 T9 differences still existdue to unknown spins and resonance strengths of several resonances. Especially after thedoublet at 6.2261 MeV and 6.2543 MeV is resolved it would be ofinterest to perform theresonance-strength measurements for these two levels, because these two levels can be thedominant contributors to the rate for the21Na(p,γ)22Mg reaction at X-ray burst tempera-tures between 0.9 T9 and 1.1 T9.

5.9 Summary

In this section we discussed measured22Mg levels and their influence on the18Ne(α,p)21Naand21Na(p,γ)22Mg reaction rates. With an unprecedented resolution of 13 keV (FWHM)for a (p,t) experiment we resolved sixty two22Mg levels, twelve of which were observedfor the first time. The errors in excitation energies for someof the measured levels de-creased by an order of magnitude. This leads to a decrease in the error in the calculatedreaction rates, which originates from the errors in the excitation energies of the resonances,to below 15%.

By measuring the 6.2261 MeV (4+) level we confirmed the suggestion by Batemanetal. [11] that the observed peak at 6.2411 MeV is a doublet. Furthermore, we resolved fourlevels in the 7.0 to 7.1 MeV excitation-energy interval in22Mg. In the22Mg excitation-energy region relevant for the18Ne(α,p)21Na reaction at stellar temperatures we measuredfor the first time a level at 8.743 MeV and four more levels above 11 MeV excitation energy.

On the basis of measured excitation energies in22Mg, spin-parity mirror assignmentsand adopted spectroscopic factors from previous works we have deduced18Ne(α,p)21Na

5.9. Summary 97

and21Na(p,γ)22Mg reaction rates. For levels with unknownα-spectroscopic factors weadopted a constantα-spectroscopic factor for the same spin value for the18Ne(α,p)21Nareaction-rate calculations. Furthermore, we included allmeasured22Mg levels above theα-emission threshold. For this reaction, our calculated rates are a factor of five largercompared to previous calculations.

The determined21Na(p,γ)22Mg reaction rates are very similar to those calculated inRef. [18]. The reason for this similarity is that the same resonance strengths were used inthe present work and that of Ref. [18]. Our work shows that an additional evaluation ofthe 6.2261 MeV and 6.2543 MeV resonance strengths would be ofinterest though it can beseen from Fig. 5.18 that this may probably not change the21Na(p,γ)22Mg reaction ratesdramatically.

Chapter 626Si data and their astrophysical implications

In this chapter we will discuss in detail the28Si(p,t)26Si data. Our data will be comparedwith previous results and possible spin assignments will begiven. All data will be includedin the25Al(p,γ)26Si and22Mg(α,p)25Al reaction-rate calculations.

The angular distributions for the g.s. 0+, 1.796 MeV 2+ and 5.512 MeV 4+ levelsobtained in the28Si(p,t)26Si reaction are presented in Fig. 6.1 together with the angu-lar distributions calculated with DWUCK4. A description ofthe input parameters for theDWBA calculations can be found in Appendix A. In this case also, we found a discrepancybetween the measured and calculated angular distributionssimilar to that found for the24Mg(p,t)22Mg reaction. Therefore, for26Si levels which have unknown spin-parity val-ues, we will rely on the mirror spin-parity assignments using information for mirror levelsin 26Mg.

6.1 26Si and its mirror nucleus 26Mg

26Si is an unstable, proton-rich nucleus with a half-life of 2.23 s. In the present28Si(p,t)26Siexperiment we obtained high-resolution26Si spectra (see next three sections). Similar tothe discussion on22Mg presented in Chapter 5 we need additional data to perform reaction-rate calculations. We were not able to obtain spin-parity information for the26Si levelsdirectly from our experiment and there are no resonance-strength data for the25Al(p,γ)26Sior 22Mg(α,p)25Al reactions. Consequently, we used the mirror nucleus of26Si, which is26Mg, as a source for missing information that is necessary to perform the calculations forthe25Al(p,γ)26Si and22Mg(α,p)25Al reactions.

26Mg is a stable nucleus with a natural isotopic abundance of 11.01%. Consequently,its nuclear structure is much better known than that of26Si. Information on levels (i.e. ex-citation energy and spin-parity) in26Mg can be found in Ref. [61]. Using proton, neutron,25Al, and 25Si nuclear masses from Ref. [53] and22Mg and26Si masses from Ref. [49],we calculated the proton, alpha, and neutron-emission thresholds in26Mg to be 14.1458MeV, 10.6148 MeV, and 11.0931 MeV, respectively.

100 6. 26Si data and their astrophysical implications

SinceΓγ values for levels in26Si are needed to calculate the25Al(p,γ)26Si reactionrates (see Section 6.5), we determined them by comparing to mirror levels in26Mg, becauseall 26Mg levels below theα-emission threshold can decay only viaγ-decay. The spin-parityvalues for levels in26Mg are known up to 9.2 MeV with a few exceptions. Therefore, thesevalues are used to assign the spin-parity values for levels in 26Si.

TheΓp values are calculated using26Mg single-particle (neutron) spectroscopic factorsfrom Ref. [69], which have been corrected with single-particle reduced widths taken fromRef. [70], see Section 6.5.

The Sα values for26Si, necessary for the22Mg(α,p)25Al reaction-rate calculations, aretaken from its mirror nucleus,26Mg, see Section 6.6. These values are calculated from thedata of Ref. [71].

0 5 10 15 20Θ

c.m. (deg)

10-3

10-2

10-1

100

dσ/d

Ω (

mb/

sr)

g.s. 0+

calculated

0 5 10 15 20Θ

c.m. (deg)

1.796 2+

calculated

0 5 10 15 20Θ

c.m. (deg)

10-3

10-2

10-1

100

5.512 4+

calculated

Figure 6.1: The measured and calculated angular distributions for the g.s. 0+, 1.796 MeV 2+ and5.512 MeV 4+ levels of26Si. See further Fig. 4.5 for more details. The solid line is used to guidethe eye.

6.2. Calibration region (g.s. - 5.5123 MeV) 101

6.2 Calibration region (g.s. - 5.5123 MeV)

In this section we will discuss levels below the proton-emission threshold. In Fig 4.10,spectra obtained at spectrometer angles−0.3, 8 and 17 are shown with calibration levelsindicated with a∗. In Fig 6.2, we display a blow-up of the same26Si spectra for theexcitation-energy region from 4 to 5.6 MeV.

The deduced energies of excited states in26Si below the proton-emission threshold arelisted in the third column of Table 6.1; previous experimental results are listed in columns4 to 10. The adopted spin-parity values for the26Si nucleus, taken from Ref. [29] are listedin the first column of Table 6.1 and the mirror assigned spin-parity values are listed in thesecond column.

0

50

100

150

0

10

20

30

0

20

40

60

4.2 4.4 4.6 4.8 5 5.2 5.4

28Si(p,t) 26Si

5.14

2

4.82

7

4.80

6

4.44

6*

14O

14O

5.51

2

4.18

64.

137

5.28

6Θ=−0.3

Θ=8

Θ=17 ο


ο

ο

Cou

nts/

2.5

keV

Figure 6.2: 26Si spectra in the region 4 to 5.6 MeV taken at spectrometer angles−0.3, 8, and 17.A resolved doublet at 4.806 MeV and 4.827 MeV can be observed.The calibration line is markedwith ∗. The determined excitation energies for26Si are listed in the third column of Table 6.1. Seefurther Fig. 4.5 for more details.

10

26.

26Sidata

andtheir


Table 6.1: The calibration region encompassing the26Si excitation energies below the proton-emission threshold, i.e. 5.5123 MeV, with the adoptedspin-parity assignments.

Jπ Jπ Present Ref. [29] Ref. [28] Ref. [27] Ref. [72] Ref. [73] Ref. [74] Ref. [46] adoptedadoptedb mirrorc (p,t) (3He,n) (3He,6He) (p,t) (3He,n) (p,t) (3He,n) compilation presentd

0+ 0+ g.s.∗ g.s.a g.s.a g.s.a g.s. g.s. g.s. g.s. g.s.2+ 2+ 1.7959∗ 1.7959a 1.7959a 1.7959a 1.800(30) 1.795(11) 1.7959(2) 1.7959(2) 1.7959(2)2+ 2+ 2.7835∗ 2.7835a 2.7835a 2.7835a 2.780(30) 2.790(12) 2.7835(4) 2.7835(4) 2.7835(4)0+ 0+ 3.3325∗ 3.3325a - 3.3325a 3.330(30) 3.339(19) 3.3325(3) 3.3325(3) 3.3325(3)- (3+) (3.749(4)) 3.756a - 3.756a 3.760(30) - 3.756(2) 3.756(2) 3.756(2)- - - - - - - - 3.842(2) 3.842(2) 3.842(2)- (4+) - - - - - - (4.093(3)) (4.093(3)) (4.093(3))2+ 2+ 4.1366(28) 4.138(4) 4.144(8) 4.155(2) 4.140(30) - 4.138(1) 4.138(1) 4.1379(9)3+ 3+ 4.186(4) 4.183(4) 4.211(16) 4.155(2) - 4.183(11) - 4.183(11) 4.1854(27)2+ 2+ 4.446∗ 4.446a 4.446a 4.445a 4.450(30) 4.457(13) 4.446(3) 4.446(3) 4.4466(29)- (4+) 4.8057(25) 4.806a 4.806a 4.805a 4.810(30) 4.821(13) 4.806(2) 4.806(2) 4.8061(16)- (0+) 4.8270(25) - - - - - - - 4.8270(25)2+ 2+ 5.1415(17)e 5.145(4) 5.140(10) 5.145(2) - - - - 5.1431(12)- - - - - - - 5.229(12) - 5.229(12) 5.229(12)4+ 4+ 5.286(6) 5.291(4) 5.291 5.291(3) 5.310(30) - - 5.330(20) 5.2903(22)4+ 4+ 5.5116(25) 5.515(4) 5.526(8) 5.515(5) - 5.562(28) - 5.562(28) 5.5139(19)

∗ Used in the present calibration.a Used for calibration in the previous articles.b Adopted Jπ values by Ref. [29].c Mirror assignments, Fig. 6.3.d Weighted average, only data given by Endt [46] are excluded.e Averaged from the spectra at magnetic-field settings B1 and B2 at spectrometer angle−0.3.

6.2. Calibration region (g.s. - 5.5123 MeV) 103

4.1366(28) MeV 2+, 4.186(4) MeV 3+: Our measured excitation energies for thisdoublet are in agreement with all experiments where this doublet is resolved, see Refs.[29, 73, 74].

4.8057(25) MeV (4+), 4.8270(25) MeV (0+): An indication that a doublet exists ataround 4.8 MeV was given by Bohneet al. [72], Iliadis et al. [26], Caggianoet al. [28]and Bardayanet al. [27]. In the present experiment where a high resolution of 13keVFWHM was achieved this doublet was resolved into levels at energies 4.8057(25) MeVand 4.8270(25) MeV. These two levels can be observed clearlyin the spectrum obtainedat a spectrometer angle of−0.3; see Fig. 6.2. The level with the lower excitation energyof 4.8057(25) MeV can very well correspond to the level measured by Bellet al. [74] at4.806(2) MeV. Possible 0+, 2+, and 4+ spin-parity values were correlated with this doubletin earlier articles [26, 27, 28, 72]. In the mirror spin-parity assignments given in Fig. 6.3we suggest spin-parity 4+ and 0+ for the 4.8057(25) MeV and 4.8270(25) MeV levels,respectively.

5.1415(17) MeV 2+, 5.5116(25) MeV 4+: In Section 4.6 we already emphasized thatwe will not use these two levels for the calibration of our26Si spectra. This was done inorder to determine their excitation energies independently, considering the fact that there isa doublet around 4.8 MeV which was treated as a single level inprevious experiments; seeRefs. [27, 28, 29]. Our values are slightly lower, but still in agreement with the previousresults. The presently adopted excitation energy of the second level is 5.5139(19) MeV,which overlaps with the26Si proton-emission threshold 5.5123(1) MeV, calculated usingthe26Si mass given by Parikhet al. [49].

The26Si level measured at an excitation energy of 5.229(12) MeV inthe (p,t) experi-ment by Paddock [73] is not observed in any other experiment,including the present (p,t)experiment and that by Bardayanet al. [27]. If the 26Mg 1+ state at 5.691 MeV corre-sponds to the26Si 1+ level at an excitation energy of 5.672(4) MeV, as suggested in Fig.6.3, then there is no corresponding26Mg mirror level to the26Si 5.229(12) MeV state.Consequently, the observed level at 5.229(12) MeV in Ref. [73] can be an overlap of the5.1415(17) MeV and 5.286(6) MeV levels and the10Cg.s. impurity at the same position,which could not be resolved by Paddock [73].

The excitation energy of the state at 5.286(6) MeV has been determined from the spec-trum measured at a spectrometer angle of 8, because this level was affected by an14Ocontaminant line at−0.3 spectrometer angle; see Fig. 6.2. In spite of the higher statisticsat a spectrometer angle of 17, we used the result at a spectrometer angle of 8 because atthis angle the systematic error is smaller than at a spectrometer angle of 17.


g.s. 0+

2.783 2+

3.332 0+

4.138 2+4.185 3+

4.446 2+

4.806 (4+)4.827 (0+)

(5.229)5.143 2+

5.290 4+

5.514 4+

g.s. 0+

1.808 2+

2.938 2+

3.589 0+

3.942 3+

4.319 4+4.333 2+4.350 3+

4.835 2+4.901 4+4.972 0+

5.292 2+

5.476 4+

5.691 1+5.715 4+

4.093 (4+)

26Si 26Mg

5.672 1+

1.796 2+

2.783 *

4.1366(28)4.186(4)

4.446*

3.332*

4.8057(25)4.8270(25)

5.286(6)

5.5116(25)

g.s. *

(3.842)3.756 (3+)(3.749(4))

5.1415(14)

adoptedenergy(MeV)

present(p,t)(MeV)

1.796 *

Figure 6.3: Possible26Si mirror assignments for levels below the proton-emissionthreshold. The26Mg spin-parity values without brackets are taken from previous experiments Ref. [61]. The26Sispin-parity values within brackets are possible mirror assignments. The full (dashed) arrows linesindicate definite (tentative) mirror assignments. The26Mg spin-parity values are from Ref. [61]. Seefurther Fig. 5.3 for more details.

6.3 Region above the proton-emission threshold (5.5123

MeV - 9.164 MeV)

In this section we will discuss26Si levels above the proton-emission threshold of 5.5123MeV. The spectrum taken at the spectrometer angle of−0.3 using the 0.70 mg/cm2 thick

6.3. Region above the proton-emission threshold (5.5123 MeV - 9.164 MeV) 105

28Si target is presented in the upper panel of Fig. 6.4. To identify the 10C and14O contam-inant levels, we show in the bottom panel of Fig. 6.4 a spectrum taken at the same angleusing a 1.00 mg/cm2 thick Mylar target.

0

20

40

60

80

100

120

140

160

180

200

0

20

40

60

80

100

120

140

5.5 6 6.5 7 7.5 8 8.5 9

28Si(p,t) 26Si

(a)

(b)

14

10 C

14 O

10 C

14 O

O

O10 O

14 O 10 C

14 O

14


Cou

nts/

5 k

eV

Figure 6.4: (a) The spectrum taken using a 0.70 mg/cm2 thick 28Si target at the spectrometer angleof −0.3. (b) The spectrum taken using a 1.00 mg/cm2 thick Mylar target at the same spectrometerangle. The10C and14O impurity lines are indicated.

The 0.7 mg/cm2 thick 28Si target1 consisted of three thin layers. In addition, we col-lected also spectra using a 1.86 mg/cm2 thick natural Si target at all three spectrometerangles. By comparing differential cross sections for the26Si levels measured at differenttime intervals on both Si targets (enriched and natural) we concluded that one layer was

1This target was made by Greene and Berg [75].


burned at the beginning of the measurements and one more later on. These burned layersdecreased the amount of collected data and this can be seen inthe 26Si spectra after sub-traction of the10C and14O contaminants; see Fig. 6.5. The achieved statistics is muchlower than achieved for the24Mg target discussed in Chapter 4.

The burning of the Si target layers introduced a problem for the subtraction of contam-inant lines from the26Si spectra. The spectra obtained with the three-times thicker Mylartarget have a resolution which is worse as compared to that ofthe 26Si spectra. This pre-vented us from obtaining a high quality subtracted spectra like the22Mg spectra shown inFig. 4.4. In Fig. 6.5, the remaining parts of the most prominent contaminant lines areindicated with “B”. Because of these problems we accepted in our analysis onlystronglypopulated26Si levels or levels which are clearly visible in at least two spectra. In Fig.6.5, the26Si levels between the proton-emission threshold andα-emission threshold areindicated by their excitation energy in the spectrum where their energies were determined.

The determined26Si excitation energies for levels between the proton-emission andα-emission thresholds are listed in column 3 of Table 6.2. Previous experimental results inthis region are listed in columns 4 - 9 of Table 6.2. The presently adopted26Si excitationenergies are listed in the last column of Table 6.2. These have been obtained as weightedaverages of all existing data listed in columns 3 to 8. Only data listed by Endt were not takenin the averages because they are averages of earlier existing data. In Table 6.1, we showedthat our results are consistent with those from the previousexperiments for levels belowthe proton-emission threshold. However, for the26Si levels above the proton-emissionthreshold our results differ from results given by Parpottas et al. [29] and from some ofthe levels given by Bardayanet al. [27]; see columns 3, 4, and 6 of Table 6.2. As alreadydiscussed in section 6.2, the differences result from the use of the doublet at 4.806 MeVfor the last calibration point in Refs. [27, 29]. However, our data are consistent with datagiven by Bohneet al. [72] and Paddock [73].


0

50

100

150

200

0

20

40

0

20

40

5.5 6 6.5 7 7.5 8 8.5 9

28Si(p,t) 26Si

Cou

nts/

10

keV

B

B

B

B B

7.4146.

454

6.37

8

6.29

4

7.14

9(7

.196

)

(6.7

83)

7.87

3

B

B

B

B B

B8.

987

7.47

7

7.52

17.

659

7.70

0

(8.6

85)

Θ=8 ο

Θ=17 ο


Θ=−0.3 ο

8.22

08.

267

B

(8.5

55)

Figure 6.5: The 28Si(p,t)26Si spectra between the proton-emission threshold and theα-emissionthreshold taken at the spectrometer angles−0.3, 8, and 17. The excitation energies of the ana-lyzed 26Si levels are marked in the spectrum where they were determined. The remaining parts ofthe subtraction of10C and14O contaminant lines are indicated by “B”. The determined excitationenergies for26Si are listed in the third column of Table 6.2. See further Fig. 4.5 for more details.

10

86.

26Sidata

andtheir


Table 6.2: Region above the proton-emission threshold (5.5123 - 9.164MeV).

Jπ Jπ Present Ref. [29] Ref. [28] Ref. [27] Ref. [72] Ref. [73] Ref. [46] adoptedadopteda mirrorb (p,t) (3He,n) (3He,6He) (p,t) (3He,n) (p,t) compilation presentc

4+ 4+ 5.5116(25) 5.515(4) 5.526(8) 5.515(5) - 5.562(28) 5.562(28) 5.5139(19)1+ 1+ - 5.670(4) 5.678(8) - - - - 5.672(4)0+ d 0+ (5.919(12))e 5.912(4) - 5.916(2) 5.910(30) - - 5.9152(18)0+ 3+ (5.942(20))e 5.946(4) 5.945(8) - - 5.960(22) 5.940(25) 5.946(4)2+ 2+ 6.2940(24) 6.312(4) - 6.300(4) 6.320(30) - - 6.2991(18)2+ (4+) 6.3778(29) 6.388(4) - 6.380(4) - 6.381(20) 6.350(25) 6.3810(20)0+ 0+ 6.4546(28) 6.471(4) - - 6.470(30) - 6.470(30) 6.4600(23)3− 3− 6.783(5) 6.788(4) - 6.787(4) 6.780(30) 6.786(29) 6.789(17) 6.7861(23)- (5+) - - - - 6.880(30) - 6.880(30) 6.88(3)- (3+) - - - 7.019(10) - - - 7.019(10)2+ 2+ 7.149(5) 7.152(4) - 7.160(10) 7.150(30) 7.150(15) 7.150(13) 7.1514(28)- (5+) (7.196(8)) - - - - - - (7.196(8))0+ 0+ 7.4135(23) 7.425(4) - 7.425(7) 7.390(30) - 7.390(30) 7.4169(19)2+ 2+ 7.477(12) 7.493(4) - 7.498(4) 7.480(30) 7.476(20) 7.489(15) 7.4941(27)- (5−) 7.521(12) - - - - - - 7.521(12)- (2+) 7.659(12) - - - - - - 7.659(13)3− 3− 7.700(12) 7.694(4) - 7.687(22) - 7.695(30) 7.695(30) 7.694(4)1− 1− 7.873(4) 7.899(4) - 7.900(22) 7.900(30) 7.902(21) 7.892(15) 7.8829(24)- (3+) - - - - 8.120(30) - 8.120(20) 8.12(3)

(1−) 8.220(5) - - - - - - 8.220(5)- (2+) 8.267(4) - - - - - - 8.267(4)- (2+) (8.555(4)) - - - - - - (8.555(4))- (4+) (8.685(12)) - - - 8.700(30) - - 8.687(11)- (4+) 8.987(7) - - - - - - 8.987(7)

a Adopted Jπ values by Ref. [29].b Mirror assignments; see Fig. 6.6.c Weighted average, only data given by Endt [46] areexcluded; see text.d Adopted from Ref. [27], in Ref. [29] a 3+ spin-parity was suggested.e Tentative excitation energies from themeasurement at the spectrometer angle of 17. These are not taken in the averaging process.


In the following we will discuss some of the levels individually.5.672(4) MeV 1+ (last column of Table 6.2): This level is assigned 1+ spin-parity in

Refs. [28, 29]. This level is not observed in the present experiment, nor in previous (p,t)experiments [27, 73]. Because of this we assumed an unnatural 1+ spin-parity for this levelin our spin-parity mirror assignments (Fig. 6.6), which is consistent with Refs. [28, 29].

(5.919(12)) MeV 0+, (5.942(20)) MeV 3+: These levels are observed in previousexperiments [27, 29, 72]. The level at 5.919(12) MeV was assigned 0+ spin-parity fromthe DWBA analysis in Ref. [27] and the level at 5.942(20) MeV was assigned spin-parity3+ from mirror assignments by us and Caggianoet al. [28]. In the present experimentwe were not able to observe these two levels at−0.3 because of a10C impurity line. Ata spectrometer angle of 8 they were not observed at all, and at 17 the statistics did notallow us to make any conclusive decision.

6.2940(24) MeV 2+, 6.3778(29) MeV (4+), 6.4546(28) MeV 0+: These three levelsare clearly observed at−0.3 and 8 spectrometer angles. Their deduced excitation en-ergies are in agreement with Refs. [27, 72, 73] but not with Ref. [29]. This discrepancymight be attributed to the use of the 4.806 MeV doublet for thecalibrations performed byRef. [29]. In our present mirror spin-parity assignments wewere not able to follow thespin-parity assignments given by Parpottaset al. [29]. Because the spin-parity assignmentfor these three levels in Refs. [27, 72] were based on a DWBA analysis we adopted a 2+

spin-parity assignment for the level at 6.2940(24) MeV. In addition we assigned a (4+)spin-parity to the level at 6.3778(29) MeV; see Fig. 6.6.

7.521(12) MeV (5−), 7.659(12) MeV (2+): We resolved these two, previously notobserved, states. The 12 keV error is caused by the larger uncertainty in the determinationof Ex at the 17 spectrometer angle, which is related to kinematic broadening. These twolevels are not clearly observed at−0.3 and 8 spectrometer angles because at those anglesthey are covered by a broad10C resonance; see Fig. 6.4.

8.220(5) MeV (1−), 8.267(4) MeV (2+), 8.987(7) MeV (4+): These three resolvedlevels are observed at all three spectrometer angles. Theirpossible mirror spin-parity as-signments can be seen in Fig. 6.6.


6.746 2+

5.229

5.290 4+

5.514 4+

5.672 1+

6.299 2+

6.880 (5+)

7.019 (3+)

7.151 2+(7.196) (5+)

7.494 2+7.521 (5−)

7.659 (2+)7.694 3−

8.120 (3+)

5.143 2+

5.476 4+

5.691 1+5.716 4+

6.125 3+

6.256 0+

6.622 4+6.634 (0 −4)+

6.876 3−

6.978 5+

7.062 1−7.100 2+

7.200 (0,1)+7.242 3+7.261 (2,3)−7.283 4−7.349 3−7.371 2+7.395 5+7.428 (0,1)+

7.541 2−

7.697 1−7.726 3+7.773 4+ 7.816 (2,3)+

7.824 3−

7.840 2+7.851

7.953 5−

8.033 2+8.053 2−8.185 3−8.201 6+8.228 1−8.251 3+

8.399 8.459 3+

8.464 (2 − 6)+

8.472 6+

8.504 1−8.532 2+8.577 1+8.625 5+8.670 (3,5)8.706 4+

8.864 2+8.903 3−8.930 4+8.959 1−9.0209.045 2−9.064 5+9.111 6+9.169 6−9.2069.239 1(+)

5.292 2+

7.677 4+

26Si 26Mg

5.5116(25)

6.2940(24)

6.3778(29)

6.4546(28)

(6.783(5))

7.149(5)

7.4135(23)

7.521(12)

7.659(12)7.700(12)

7.873(4)

8.267(4)

6.460 0+

6.786 3−

8.220 (1−)8.220(5)

(8.685(12))

9.261 4+

5.915 0+ (a)

5.1415(14)

5.286(6)

(5.919(12))(5.942(20))

7.477(12)

5.946 3+

present(p,t)(MeV)

adoptedenergy(MeV)

6.381 (4+) (a)

(7.196(8))

7.417 0+

(8.555(4))

8.687 (4+)

8.267 (2+)

7.883 1−

8.987 (4+)

(8.555) (2+)

8.987(7)

Figure 6.6: The possible26Si mirror assignments for levels above the proton-emissionthreshold.The 26Si spin-parity assignments without brackets are taken fromprevious experiments [61]. The26Si spin-parity values within brackets are possible mirror assignments. The full (dashed) arrowsindicate definite (tentative) mirror assignments. The22Ne spin-parity values are from Ref. [61]. The26Si adopted energies are taken from the last column of Table 6.2. These spin-parity assignments willbe used for reaction-rate calculations. (a) From DWBA analysis in Ref. [27]. See further Fig. 5.3 formore details.

6.4. Region above the α-emission threshold (9.164 MeV) 111

6.4 Region above theα-emission threshold (9.164 MeV)

The26Si levels above theα-emission threshold are important for the22Mg(α,p)25Al reac-tion in X-ray bursts and supernovae; see Section 2.5. In the present experiment we iden-tified four 26Si levels above theα-emission threshold and for three more levels tentativeexcitation energies were determined. All these levels are listed in Table 6.3 and shown inFig. 6.7. Possible mirror assignments are given in column 1 of Table 6.3.

The tentative levels above 10.0 MeV are listed because they could be identified at twospectrometer angles at a consistent excitation energy. Because of low statistics and a stronginfluence of the10C and14O impurity lines, we have not been able to unambiguouslyidentify levels above 10 MeV excitation energy in26Si. Large errors due to the subtractionof impurity lines are present. We also fitted the area betwen 10 and 12 MeV with a constantstraight line and obtained a reduced chi-square of around unity. Consequently, all listedpossible levels above 10 MeV excitation energy can be simplystatistical fluctuations.

Table 6.3: Region above theα-emission threshold (9.164 MeV) in26Si.

Jπ Ex26Si Ex

26Mgmirror (MeV) (MeV)(4+) 9.314(6) 9.579(3)(2+) 9.604(13) 9.85652(6)(5−) 9.760(4) 10.040(2)(0+) 9.9017(25) 10.159(3)- (10.434(13)) -- (10.65(6)) -- (11.01(4)) -

There are many more observed levels in26Mg for this excitation-energy range as com-pared to26Si, and there is no clear marking point for mirror assignments. Because of thisa mirror energy difference between 250 keV and 350 keV was maintained as for levelsat excitation energy higher than 7.5 MeV as shown in Fig. 6.6.Because the (p,t) reac-tion preferably excites natural-parity states, we correlated observed26Si levels with knownnatural-parity states in26Mg. In this way, we obtained mirror assignments as presentedinTable 6.3.

6.5 Astrophysical implications for the 25Al(p,γ)26Si reac-

tion

The 26Si levels relevant for the25Al(p,γ)26Si reaction have been studied in recent exper-iments [26, 27, 28, 29]. The difference between the excitation energies measured in the


28Si(p,t) 26SiC

ount

s/ 2

0 ke

V


Θ=−0.3 ο

Θ=8 ο

Θ=17 ο

9.76

0

9.90

2

9.31

4

(10.

650)

9.60

4

(10.

434)

(11.

01)

0

100

200

300

0

100

200

0

50

100

150

9 9.5 10 10.5 11 11.5 12

Figure 6.7: The 28Si(p,t)26Si spectra above theα-emission threshold measured at−0.3, 8, and17 spectrometer angles. The energy determined for each peak ismarked in the spectrum used todetermine it. The determined excitation energies for26Si are listed in the second column of Table6.3. See further Fig. 4.5 for more details.

present experiment and those in Ref. [29] for levels above the proton-emission thresholdhas already been discussed in Section 6.3. For the discussion of the 25Al(p,γ)26Si and22Mg(α,p)25Al reaction rates we will use only adopted excitation energies listed in the lastcolumns of Table 6.2 and Table 6.3, respectively.

From Refs. [26, 27, 28, 29] it can bee seen that the resonancesat 0.159(4) MeV (1+),0.4029(18) MeV (0+), and 0.434(4) MeV (3+) dominate the25Al(p,γ)26Si reaction ratesfor all astrophysically relevant temperatures. In the present experiment we did not observe

6.5. Astrophysical implications for the 25Al(p,γ)26Si reaction 113

any of these three levels, including the natural-parity 0+ level. Taking into account possi-ble errors introduced in previous experiments by using the doublet at 4.806 MeV for thecalibration of the excitation energy, a new measurement of the excitation energies for thesethree levels will be valuable.

On basis of the adopted26Si excitation energies, listed in the last column of Table 6.2,we have calculated the25Al(p,γ)26Si reaction rates. The important difference betweenthe present calculations and those done previously [28, 29]is a new value for the proton-emission threshold for26Si. Instead of the value previously used (5.518 MeV), the new26Si proton-emission threshold value is 5.5123(10) MeV; see the conclusion in Section 4.6.

As a consequence the 5.5139(19) MeV level is actually above the26Si proton-emissionthreshold. However, its 0.0016 MeV resonance energy is wellbelow the Gamow windowfor the 25Al(p,γ)26Si reaction for temperatures above 0.01 T9. Therefore, we will not usethis level in our reaction-rate calculations.

For temperatures between 0.01 T9 and 1.5 T9 the contribution of the direct reaction ratein the25Al(p,γ)26Si reaction is taken from Ref. [26].

Since there is no direct measurement of26Si resonance strengths we rely on the calcu-lations using Eq. 2.28. The first few26Si levels above the proton-emission threshold arewell below the Coulomb barrier and, consequently,Γp ≪ Γγ andωγ ≈ ωΓp. The protonpartial decay width is calculated using the recipe outlinedin Ref. [70]:

Γp =3~

2

µR2n

PlΘ2spC

2S (6.1)

whereΘ2sp is the dimensionless single-particle reduced width, and the rest of the parameters

are the same as for Eq. 2.29. In order to be consistent with theformalism given in Chapter2, theΘ2

sp is given by:

Θ2sp =

Rn

3φ2

l (Rn) (6.2)

and not as given in Ref. [70]:

Θ2sp =

Rn

2φ2

l . (6.3)

Therefore, in order to use theΘ2sp values calculated according to Iliadis [70] we will scale

them with a factor23 . Theφ2l (Rn) denotes the square of the single-particle radial wave

function of thel orbit at the interaction radiusRn given by Eq. 2.9. In the literature thesingle-particle reduced widths are often set equal to unity. As shown by Iliadis [70] thisapproach can produce a significant error in theΓp calculations. Iliadis investigated thesystematic dependence ofΘ2

sp on variations in bombarding energy, target mass, charge,interaction radius, and radial and angular quantum numbers. We calculated the protonpartial reduced widths for the26Si resonances by using the formalism given in Ref. [70].


In Table 6.4 we list levels in26Si above the proton-emission threshold with their protonand gamma partial decay widths. The proton partial decay widths marked with superscripta in Table 6.4 are calculated on basis of the26Mg single-particle spectroscopic factors takenfrom Ref. [69]. The theoretically calculated single-particle spectroscopic factor taken fromRef. [26] is marked with superscriptb in Table 6.4. The single-particle reduced widthsΘ2

sp

are calculated on basis of data given by Iliadis [70]. The penetrability of the Coulomb andcentrifugal barrier was calculated using the PENE code [35].

In order to calculateΓγ , we used the measured half-life data for decay viaγ-emission[61] of corresponding levels in the mirror nucleus26Mg. As for the case of the22Mg data(Section 5.8) we corrected in the same way these values for the difference inγ-ray transi-tion energies of mirror transitions; those levels are marked by the superscriptc in Table 6.4.

Table 6.4: The 25Al(p,γ)26Si reaction: resonance energies, spin assignments and resonancestrengths.

Exf Eres Jπ Γp Γγ ωγ

(MeV) (MeV) mirror (eV) mirror (eV) (eV)5.6716 0.1593 1+ 4.22E-8a 1.10E-1e 1.06E-85.9152 0.4029 0+ 1.05E-2b 8.03E-3c 3.79E-45.9462 0.4339 3+ 7.93E-1a 4.58E-2c 2.53E-26.2991 0.7868 2+ 3.50E+0a 2.65E-2c 1.10E-26.3810 0.8687 (4+) 1.51E+0b 2.65E-2c 1.96E-26.4600 0.9477 0+ - 9.04E-2c 7.54E-36.7861 1.2738 3− 7.70E+2a 7.66E-3c 4.47E-36.8800 1.3677 (5+) 1.28E+2a 5.39E-2c 4.94E-27.0190 1.5067 (3+) 2.64E+3a 9.32E-2c 5.44E-27.1514 1.6391 2+ - 1.00E-2d 4.17E-37.1955 1.6832 (5+) 9.23E+2a 4.53E-2c 4.15E-27.4169 1.9046 0+ - 1.00E-2d 8.33E-47.4941 1.9818 2+ - 1.00E-2d 4.17E-37.5205 2.0082 (5−) - 3.26E-2c 2.99E-27.6592 2.1469 (2+) - 1.00E-2d 4.17E-37.6944 2.1821 3− - 1.00E-2d 5.83E-37.8829 2.3706 1− - 5.17E-1c 1.29E-18.1200 2.6077 (3+) 1.91E+4a 1.00E-2d 5.83E-3

a Particle decay widths calculated using the single-particle spectroscopic factors from themirror nucleus26Mg, Ref. [69].b Theoretically calculated single-particle spectroscopicfactor taken from Ref. [26].c Γγ values calculated from the half-life data for26Mg mirror levels [61].d A constant value of 0.01 eV is assumed forΓγ .e Theoretically calculatedΓγ value, taken from Ref. [26].f 26Si excitation energies are taken from last column in Table 6.2.Jπ values without brackets are taken from Ref. [29].Jπ values within brackets are obtained by the mirror assignments given in Fig. 6.6.

6.5. Astrophysical implications for the 25Al(p,γ)26Si reaction 115

0.01 0.1 1T

9 (K)

10-50

10-45

10-40

10-35

10-30

10-25

10-20

10-15

10-10

10-5

100

Rea

ctio

n ra

tes

(cm

3 m

ol-1

s-1

)

0.1593 MeV 1+

0.4029 MeV 0+

0.4339 MeV 3+

DC

25Al(p,γ)26

Si

Figure 6.8: The contributions of the first three resonances above the proton-emission threshold tothe25Al(p,γ)26Si reaction rates. The direct capture rates are taken from Ref. [26].

For levels indicated by the superscriptd we adopted a constantΓγ value of 0.01 eV, be-cause there are no experimental or theoretical data for those levels. For the 0.159 MeV 1+

resonance the theoretically calculated [69]Γγ value is used, since the experimental valuehas only a lower limit [61] of 0.08 eV.

For those levels for which theΓγ andΓp values are known we calculated the resonancestrength using the exact formula given in Eq. 2.28. For thoselevels for which there is noinformation onΓp, we used Eq. 5.3, since for all levels above 5.9462 MeVΓp ≫ Γγ .

On basis of the data presented in Table 6.4, we have computed new25Al(p,γ)26Si reac-tion rates. The contributions from the first three resonances and direct capture are presentedin Fig. 6.8. The resonant reactions dominate for temperatures beyond 0.04 T9.

As can be seen from Fig. 6.8, the resonance reaction rates arecompletely dominatedby the two unnatural parity resonances (0.159 MeV 1+ and 0.434 MeV 3+). This is in con-tradiction with the results shown in Ref. [29] where the 0.4029 MeV resonance dominatesfor stellar temperatures above 1 T9.

In Fig. 6.9 we show the total25Al(p,γ)26Si reaction rates in comparison with thosecomputed by Caggianoet al. [28] and Parpottaset al. [29]. The main difference betweenthe present calculations and those listed in Refs. [28, 29] is in the temperature range from0.04 T9 up to 0.2 T9. This difference is due to using different values for the single-particlespectroscopic factors. In the present calculations the single-particle spectroscopic factorshave been taken from Ref. [69] and in the previous calculations they are taken from Il-


0.1 1T

9 (K)

10-25

10-20

10-15

10-10

10-5

100

Rea

ctio

n ra

tes

(cm

3 m

ol-1

s-1

)

presentCaggiano et al. [28]Parpottas et al. [29]

25Al(p,γ)26

Si

Figure 6.9: The 25Al(p,γ)26Si reaction rates as function of temperature. The dotted anddashedcurves indicate reaction rates calculated in Refs. [28, 29], respectively.

iadis et al. [26]. At stellar temperatures above 0.2 T9 all calculated reaction rates are inagreement.

6.6 Astrophysical implications for the22Mg(α,p)25Al reac-

tion

In Section 1.4 we mentioned that the26Si levels above theα-emission threshold are im-portant for the bolometrically double-peaked type I X-ray bursts. 22Mg is aβ+-unstablenucleus which has a lowQ-value for the22Mg(p,γ)23Al reaction. Consequently, photo-disintegration of23Al prevents a significant flow through a subsequent23Al(p,γ) reaction.However, this waiting point can be bridged by the22Mg(α,p)25Al reaction. From Section6.4 we know that there are no published experimental data forthe relevant region in26Si.Since we could not make spin assignments for the states identified in the present experi-ment we will rely on the mirror spin assignments given in Section. 6.4. The required alphaspectroscopic factors, which are used to perform the reaction-rate calculations are obtainedfrom Ref. [71] for the26Mg levels. The penetrability through the Coulomb and centrifugalbarriers is calculated using the PENE code [35]. The obtained Sα values for26Mg are listedin Table 6.5.

6.6. Astrophysical implications for the 22Mg(α,p)25Al reaction 117

Table 6.5: The Sα parameters for the22Ne(α,n)25Mg reaction taken from Ref. [71].

Ex(26Mg) Eres Jπ Sα

(MeV) (MeV)10.693 0.078 4+ 1.49E-210.945 0.330 2+, 3− 3.71E-211.112 0.497 2+ 3.51E-311.153 0.538 1− 7.24E-311.163 0.548 2+ 3.51E-311.171 0.556 2+ 3.54E-311.183 0.568 1− 3.62E-311.194 0.579 2+ 3.46E-311.274 0.659 2+ 3.53E-311.286 0.671 1− 3.64E-311.310 0.695 1− 2.91E-211.326 0.711 1− 2.91E-211.328 0.713 1− 1.14E-1

Table 6.6: The adopted Sα values for the relevant levels in26Si.

Jπ Sα

0+ 0.0371− 0.0072+ 0.0373− 0.0074+ 0.0155− 0.007

From these Sα values we determined Sα values for26Si. From Table 6.5 it can be seenthat we have Sα values only for 1−, 2+, and 4+ natural-parity levels. For 0+ levels wewill adopt the Sα value 3.7E-2 as for the 2+ state at 10.945 MeV. For 3− and 5− stateswe adopted the value 7E-3 as for the 1− state at 11.153 MeV, because no other negativenatural-parity 3−, 5− states were observed for which we obtained an Sα value. We chosethese particular values for Sα because we observed only a few26Si levels just above theα-emission threshold, and we chose to use these Sα values for the corresponding26Mg mirrorlevels just above theα-emission threshold. In Section 6.4 we mentioned that we will onlyuse natural-parity states for the present mirror assignments. The adopted Sα values for26Siare listed in Table 6.6.

We used the method of narrow resonances to calculate the22Mg(α,p)25Mg reactionrates; the parameters are listed in Table 6.7.

As in Section 5.7 we will perform additional calculations using random spin-parityassignments to investigate the estimated error which may originate from the chosen spinassignment. In this case we will limit the spin value to 5. Thereason is that in our exper-iment we did not observe any known level with a spin value of 6 or higher. Furthermore,


Table 6.7: The adopted Sα values for the22Mg(α,p)25Al reaction.

Ex(26Si) Er Jπ Sα

(MeV) (MeV)9.314 0.150 (4+) 1.5E-29.603 0.440 (2+) 3.7E-29.760 0.596 (5−) 7.0E-39.902 0.738 (0+) 3.7E-2

Table 6.8: The spin values and resonance strengths for the four resonances in the22Mg(α,p)25Alreaction.

mirror a RND1 b RND2 b

Eres (MeV) J ωγ (eV) J ωγ (eV) J ωγ (eV)

0.150 4 1.00E-36 1 4.31E-35 4 1.00E-360.440 2 2.40E-14 1 9.41E-15 0 3.12E-140.596 5 1.27E-13 5 1.27E-13 2 2.01E-100.738 0 8.82E-08 0 8.82E-08 1 2.73E-08

a Spin and resonance strength for the mirror assignments given in Figs. 5.9 and 5.12.b Spin and resonance strength for the randomly generated spins of states.

an equal probability for all possible spin-values is assumed. The results of the calculationsare listed in Table 6.8 and shown in Fig. 6.10.

The obtained reaction rates differ by up to a factor of three,but it must be emphasizedthat the difference is small compared to calculations whereeither very small or very largespin values are used for all four levels. This is due to a largedifference in the penetrabilityfor different angular momenta. For example, the ratio between the resonance strengths forstates with angular momental=0 andl=5 can be up to 1000. A conservative calculationwould be if we use only the spin values 2 and 3, because these values are in between thesmallest and largest possible values. With the assumption that the givenα-spectroscopicfactors are correct and the correct mirror spin assignmentshave been used, then the cal-culated error only originates from the error in the excitation energy. The relative error inthe reaction rate for the interval of stellar temperatures between 0.1 T9 and 10 T9 is below17%.

From this discussion it can be concluded that the largest contribution to the uncertaintyin the calculated rates is due to the unknown values for the spin and parity and not to theuncertainty in the deduced excitation energy. Furthermore, the uncertainty introduced bythe spin-parity assignments is also larger than the uncertainty introduced by the Sα valuesused.

The contribution from all four resonances is shown in Fig. 6.11. The 0.440 MeV and

6.6. Astrophysical implications for the 22Mg(α,p)25Al reaction 119

0.1 1 10T

9 (K)

10-30

10-25

10-20

10-15

10-10

10-5

100

Rea

ctio

n ra

tes

(cm

3 m

ol-1

s-1

)

presentRND1RND2

22Mg(α,p)

25Al

Figure 6.10: The22Mg(α,p)25Al reaction rate as function of the temperature for the threedifferentcases listed in Table 6.8.

0.738 MeV resonances yield the dominant contribution for the entire interval of X-ray bursttemperatures (0.1 - 10 T9). The 0.150 MeV resonance is below the Gamow window forthe22Mg(α,p)25Al reaction, and regardless of the choice for the spin value its contributioncan not exceed the yield of the 0.440 MeV resonance. With the adopted spin value for the0.738 MeV resonance, it completely dominates the reaction rate for temperature above 0.2T9. However, if the 0.596 MeV resonance has spin 0, this resonance will dominate for thetemperature above 0.2 T9.

From the previous discussion we conclude that for a more accurate reaction-rate calcu-lation of the22Mg(α,p)25Al reaction it is necessary to obtain resonance strengths and spininformation for26Si levels above theα-emission threshold. Since the28Si(p,t)26Si reactionpreferably excites natural-parity states, it is necessaryto perform experiments where alsounnatural-parity states can be studied, e.g.29Si(3He,6He)26Si. The large contribution ofunnatural-parity states can be seen in Section 6.5, where wediscussed the25Al(p,γ)26Sireaction rates.


0.1 1 10T

9 (K)

10-40

10-30

10-20

10-10

100

Rea

ctio

n ra

tes

(cm

3 m

ol-1

s-1

)

0.150 (4+)

0.440 (2+)

0.596 (5-)

0.738 (0+)

22Mg(α,p)

25Al

Figure 6.11: The22Mg(α,p)25Al reaction rate as function of temperature with the separate contribu-tions from all four resolved resonances. See further Fig. 4.5 for more details.

6.7 Summary

In this chapter we discussed measured26Si levels and their influence on the calculated ratesfor the 22Mg(α,p)25Al and 25Al(p,γ)26Si reactions. With an unprecedented resolution of13 keV (FWHM) for the (p,t) experiment, we were able to resolve 38 levels in26Si, 14of which were observed for the first time. The errors in excitation energies for some ofthe measured levels decreased by 50%. This leads to a decrease in the relative error in thecalculated reaction rate, which originates from the errorsin the excitation energies of theresonances, to below 17%.

By measuring the levels at 4.8056(20) MeV and 4.8270(19) MeVwe resolved a doubletat 4.806 MeV which was hitherto unresolved in previous26Si experiments. The unresolvedcombined peak was until now used as a calibration point for the determination of the exci-tation energies of higher lying levels. In the present experiment we measured for the firsttime four levels above theα-emission threshold.

Our calculated25Al(p,γ)26Si reaction rate is similar to those calculated in Refs. [28,29]. This similarity can be attributed to the use of the same proton spectroscopic factors inall calculations as well as resonance energies which are notvery much different. The maindifference between the present calculations and those listed in Refs. [28, 29] is found inthe temperature range from 0.04 T9 up to 0.2 T9. This difference is due to using differentvalues for the single-particle spectroscopic factors. At stellar temperatures above 0.2 T9 allcalculated reaction rates are in agreement.

6.7. Summary 121

Our calculated reaction rate for the22Mg(α,p)25Al reaction is for the first time calcu-lated on basis of the experimentally measured26Si levels above theα-emission threshold.

Chapter 7

Summary and conclusion

The aim of this study was to investigate the nuclear structure of 22Mg and26Si. These twonuclei play a crucial role in stellar reaction processes at high temperatures, which occur,e.g., in X-ray bursts and nova explosions. Using the nuclear structure information obtainedfor these two nuclei, we have calculated the stellar reaction rates for the following reactions:18Ne(α,p)21Na,21Na(p,γ)22Mg, 22Mg(α,p)25Al, and25Al(p,γ)26Si.

The experimental study of the24Mg(p,t)22Mg and28Si(p,t)26Si reactions has been per-formed using the Grand Raiden spectrometer of the Research Center of Nuclear Physics(RNCP), Osaka, using a 100 MeV proton beam. Taking advantageof the new dispersion-matched WS beam line, we obtained the unprecedented energy resolution of 13 keV (FWHM)for the 24Mg(p,t) and28Si(p,t)26Si reactions. This excellent resolution allowed us to re-solve 62 levels in22Mg and 38 levels in26Si. From these levels 12 and 14 levels wereresolved for the first time in22Mg and26Si, respectively. Most of these levels are locatedin astrophysically important excitation-energy regions.The importance of high-resolutionspectrometry can be judged from our measured energy spectrum for 26Si, where we couldclearly separate the 4.8056(25) MeV and 4.8270(25) MeV levels, which were hitherto un-resolved in previous26Si experiments. The unresolved combined peak was until now usedas a calibration point for the determination of the excitation energy of higher lying levels.

The calibration of the energy spectra obtained for the24Mg(p,t)22Mg and28Si(p,t)26Si reactions was performed using high resolution22Mg γ-spectrometry datafrom Ref. [16]. With these new data available for the low-lying states below the proton-emission threshold, we could avoid a systematical error introduced by the data listed byEndt [46]. In the present work we reduced the uncertainty forthe determination of theexcitation energy of levels in22Mg and26Si to 1 keV and 2 keV1, respectively. This goodaccuracy allowed us in particular to decrease the uncertainty for the calculated rate for the18Ne(α,p)21Na reaction to less than 15%.

Other important parameters for the rate calculations are the spins, parities and reso-nance strengths of particular resonances. Until now, thereare only two direct measure-ments of the resonance strengths for excited states in22Mg. Both studies have been per-

1For some of the weak levels the accuracy is much less.

124 7. Summary and conclusion

formed using a secondary ion beam; the first at TRIUMF for the21Na(p,γ)22Mg reaction[17, 18, 19], and the second at Louvain-la-Neuve for the18Ne(α,p)21Na reaction [20, 21].As was mentioned in Section 5.1, we did not succeed to obtain unique spin-parity valuesfor excited states in22Mg and26Si. These missing parameters have been inferred from thewell-known structure of mirror nuclei22Ne and26Mg, respectively. We need to mentionthat this method carries a substantial uncertainty associated with such mirror assignments.Nevertheless, our analysis provides a big step forward in the rate calculation of the relevantnuclear processes.

With accurate determination of the excitation energies of states in22Mg and26Si wedecreased the error in calculating the stellar reaction rates originating from the error in theexcitation energy. Furthermore, our data provide an essential calibration point for the deter-mination of the excitation energy of states populated by reactions using unstable ion beams.The direct reaction mechanism of the (p,t) reaction at the present bombarding energies leadsto predominant population of natural-parity levels. Consequently, it is necessary to performother experiments as12C(16O,6He)22Mg and29Si(3He,6He)26Si in order to complete theinformation on the level structure of22Mg and26Si, respectively. For example the impor-tant role of unnatural parity states has already been shown for the25Al(p,γ)26Si reaction,where the main contributions are coming from the 1+ and 3+ resonances.

This work presents the first results ever for the reaction rate of the important22Mg(α,p)-25Al reaction, based on experimental data. Our calculated rates for the21Na(p,γ)22Mg and25Al(p,γ)26Si reactions are similar to previous results [18, 28, 29]. This similarity canbe attributed to the use of the same proton spectroscopic factors in all calculations andresonance energies which are not very much different. In contrast, our calculated ratefor the 18Ne(α,p)21Na reaction is up to a factor of 5 times larger than previous results[12, 20, 21, 64]. The difference comes from the newly measured 22Mg levels above theα-emission threshold.

The direct measurements of the rate for the18Ne(α,p)21Na,22Mg(α,p)25Al and25Al(p,γ)-26Si reactions, at excitation energies up to 2 MeV above theα-emission thresholds, has notbeen possible until today because high-quality and relatively intense radioactive beams of22Mg, 25Al, and 18Ne have not yet been produced. The production of high intensity ra-dioactive beams needs to be accompanied with improvement ofα-detection techniques.This development is required to measure the weakα-branching ratio which is difficult be-cause of the small kinetic energies of the emittedα particles.

An experimental technique, which can be used for the determination of branching ratiosfor the proton-decay andα-decay of excited states in unstable nuclei has been describedby Davidset al. [6, 15]. The same technique can be used for the study of the decay ofexcited states in22Mg and26Si above theα-emission threshold. Also in this case, the (p,t)reaction can be performed in inverse kinematics to populatestates in22Mg and26Si abovetheα-emission threshold. The outgoing tritons and22Mg or 26Si recoils can be momentum-analyzed in a single magnetic spectrometer. For example, the coincidence measurement of

125

the tritons and the residual nuclei21Na and18Ne after proton-emission andα-emission,respectively, can be performed.

Appendix A

DWBA and CC calculations

For the DWBA and CC calculations performed with the codes DWUCK4 [54] and CHUCK3[55], respectively, we used proton optical-model potentials derived from the global opticalmodel potentials (OMPs) given by Ref. [56]. In the followingwe will follow the defini-tions given by Koning and Delaroche [56]. The phenomenological OMP for proton-nucleusscattering is defined as [56]:U(r, E) = −VV (r, E) − iWV (r, E) − iWD(r, E) + VSO(r, E)

−→l · −→σ

+ iWSO(r, E)−→l · −→σ + VC(r),

whereVV , VSO, WV , WD, WSO are the real and imaginary components of the volume-central (V ), surface central (D) and spin-orbit (SO) potentials, respectively.E is the labo-ratory energy of the incident particle in MeV. The various terms are given by

VV (r, E) = VV (E)f(r, RV , aV ),WV (r, E) = WV (E)f(r, RV , aV ),WD(r, E) = −4aDWD(E) d

drf(r, RD, aD),

VSO(r, E) = VSO(E)(

ℏ

mπc

)21r

ddrf(r, RSO, aSO),

WSO(r, E) = WSO(E)(

ℏ

mπc

)21r

ddrf(r, RSO, aSO),

where the form factorf(r, Ri, ai) has a Woods-Saxon shape:f(r, Ri, ai) = (1 + exp[(r − Ri)/ai])

−1,and where:A is the atomic mass number,Ri = riA

1/3 is the radius,ai is the diffuseness,VC = Zze2

2RC

(

3 − r2

R2C

)

is the Coulomb term forr 6 RC ,

VC = Zze2

r is the Coulomb term forr > RC ,Z andz are the elementary charges of target and projectile, respectively.RC = rCA1/3 is the Coulomb radius.

The potential depth parameters and Fermi energy for the global OMP are taken fromRef. [56] and have been calculated for24Mg and28Si separately when these parametersare dependent onA andZ. These are listed in Table A.1.

128 A. DWBA and CC calculations

Table A.1: Potential depth parameters and Fermi energy for the proton optical model potentials; seeRef. [56].

proton OMP 24Mg 28Si unitvp1 = 59.30 + 21.0(N − Z)/A − 0.024A 5.87×10+1 5.86×10+1 (MeV)

vp2 = 0.007067 + 4.23 × 10−6A 7.17×10−3 7.19×10−3 (MeV−1)

vp3 = 1.729 × 10−5 + 1.136 × 10−8A 1.76×10−5 1.76×10−5 (MeV−2)

vp4 = 7 × 10−9 7.00×10−9 7.00×10−9 (MeV−3)

wp1 = 14.667 + 0.009629A 1.49×10+1 1.49×10+1 (MeV)

wp2 = 73.55 + 0.0795A 7.55×10+1 7.58×10+1 (MeV)

dp1 = 16.0 + 16.0(N − Z)/A 1.60×10+1 1.60×10+1 (MeV)

dp2 = 0.0180 + 0.003802/(1 + exp((A − 156)/8)) 2.18×10−2 2.18×10−2 (MeV−1)

dp3 = 11.5 1.15×10+1 1.15×10+1 (MeV)

vpso1 = 5.922 + 0.0030A 5.99×10+0 6.01×10+0 (MeV)

vpso2 = 0.0040 4.00×10−3 4.00×10−3 (MeV−1)

wpso1 = −3.1 -3.10×10+0 -3.10×10+0 (MeV)

wpso2 = 160 1.60×10+2 1.60×10+2 (MeV)

Epf = −8.4075 + 0.01378A -8.08×10+0 -8.02×10+0 (MeV

VC = 1.73ZA−1/3/rC 5.34×10+0 6.02×10+0 (MeV)

The parameters of the global proton OMP are given by:VV (E) = vp

1 [1 − vp2(E − Ep

f ) + vp3(E − Ep

f )2 − vp4(E − Ep

f )3]

+ VC · vp1 [vp

2 − 2vp3(E − Ep

f ) + 3vp4(E − Ep

f )2],

WV (E) = wp1

(E−Epf )2

(E−Epf )2+(wp

2)2

,

rV = 1.3039− 0.4054A−1/3,aV = 0.6778− 1.487× 10−4A,

WD(E) = dp1

(E−Epf )2

(E−Epf)2+(dp

3)2

exp[−dp2(E − Ep

f )],

rD = 1.3424− 0.01585A1/3,aD = 0.5187 + 5.205 × 10−4A,VSO(E) = vp

so1exp[−vpso2(E − Ep

f )],

WSO(E) = wpso1

(E−Epf )2

(E−Epf )2+(wp

so2)2

,

rSO = 1.1854− 0.647A−1/3,aSO = 0.59,rC = 1.198 + 0.697A−2/3 + 12.944A−5/3,

The calculated phenomenological OMPs for24Mg and28Si at a proton bombarding en-ergy of 98.7 MeV, using the global proton OMP parameters given above and the parametersgiven in Table A.1, are listed in Table A.2.

For the outgoing channel the parameterization of Ref. [57] was used which is based onthe analysis of the26Mg(3He,t)26Al reaction and the parameters listed in Table A.3.

129

Table A.2: The global proton OMPs for24Mg and28Si.

24Mg 28Si unitVv 26.1810 26.2236 (MeV)Wv 9.9359 9.9302 (MeV)rV 1.1634 1.1704 (fm)aV 0.6742 0.6736 (fm)WD 1.5421 1.5439 (MeV)rD 1.2967 1.2943 (fm)aD 0.5312 0.5333 (fm)VSO 3.9105 3.9192 (MeV)WSO -0.9552 -0.9545 (MeV)rSO 0.9611 0.9723 (fm)aSO 0.5900 0.5900 (fm)rC 1.3468 1.3239 (fm)

Table A.3: Triton optical-model parameters of26Al are used for22Mg and26Si.

26AlVv 98.2 (MeV)Wv 16.5 (MeV)WD -rv 1.14 (fm)rw 1.60 (fm)av 0.85 (fm)aw 0.83 (fm)rC 1.25 (fm)

In both DWBA and CC calculations a finite-range correction parameter for the (p,t)reaction of 0.69 was used. An r.m.s. radius of 1.70 fm was usedfor the triton. In theCC calculations the followingβ2-values of the quadrupole deformation parameters wereused for28Si, 26Si, 24Mg, 22Mg: -0.398, -0.434, 0.478, 0.475, respectively. These weredetermined from the transition rates and quadrupole moments compiled in the Nuclear Datasheets by using the implicit folding model procedure [76].

Nederlandse samenvatting en vooruitblik

Nucleaire astrofysica houdt zich onder andere bezig met hetonderzoek naar het opwek-ken van energie in sterren. Daarmee samenhangend, onderzoekt men in dit vakgebied hetontstaan van de chemische elementen en hun relatieve abundantie. Sinds de ontdekking in1952 door Merrill [1] van het radioactieve element technecium in de spectra van S-sterren,is het duidelijk geworden dat inderdaad door middel van kernfusie zwaardere elementen inhet binnenste van een ster ontstaan. Omdat de halfwaardetijd van het ontdekte techneci-um tussen de 105 en 106 jaar ligt, hetgeen zeer kort is op een kosmologische tijdsschaal,betekent dit dat ook tegenwoordig astrofysische kernfusieprocessen plaats vinden.

De theorie van de oerknal voorspelt dat vrijwel alle materie, die zeer kort na de oerknalontstaan is, bestaat uit protonen en4He kernen. De zwaarste kernen, die gemaakt zijnkort na de oerknal en in nog aantoonbare hoeveelheden waargenomen kunnen worden,zijn kernen van7Li. Alle zwaardere kernen zijn ontstaan in sterren; zie referentie [2].De zogenaamde hoofdreeks sterren, die de meerderheid vormen van alle waargenomensterren, fuseren waterstof in helium. De verbranding van waterstof verloopt zeer geleidelijkin het binnenste van een ster bij een hoge temperatuur en druk. Daarbij bestaat in dester een hydrodynamisch evenwicht tussen de naar de binnengerichte zwaartekracht ende naar buitengerichte stralingsdruk. Op het moment dat alle waterstof in het binnenstevan een ster verbruikt is, zal de kern van een ster gaan krimpen onder invloed van deafnemende stralingsdruk en de vooralsnog gelijkblijvendezwaartekracht. Hierdoor zullende temperatuur en de druk in het binnenste van de ster verder stijgen. Onder bepaaldeomstandigheden, die afhangen van de massa van de ster, kan deverbranding van heliumgaan beginnen, waardoor koolstof en zuurstof worden geproduceerd. Als ook helium nietmeer ter beschikking staat voor verbranding, zal het samentrekken van het binnenste vande ster weer doorgaan, zodat vervolgens ook koolstof en zuurstof verbranding kan gaan

132 Nederlandse samenvatting en vooruitblik

optreden, waardoor nog zwaardere kernen worden gemaakt.22Na en26Al zijn twee langlevende radioactieve kernen, die na verloop van tijd verval-

len viaβ+ verval, gevolgd door het uitzenden vanγ-straling in de dochterkernen22Ne enrespectievelijk26Mg. Tijdens explosieve verbrandingsprocessen, zoals die optreden tijdensnovae explosies en zogenaamde X-ray bursts, kunnen deze kernen weggeblazen worden inde interstellaire ruimte en het verval, waarbijγ-straling wordt uitgezonden, kan wordenwaargenomen door satellieten, die richtingsgevoelige detectoren voorγ-straling aan boordhebben.

Het doel van de huidige studie was het onderzoek naar de kernstructuur van22Mgen 26Si. Beide kernen spelen een zeer belangrijke rol in astrofysische kernreacties, dieplaatsvinden bij een zeer hoge temperatuur, zoals deze voorkomt tijdens X-ray bursts ennovae explosies. Door gebruik te maken van de verkregen kennis van de structuur van dezekernen, hebben we de astrofysische reactiesnelheid berekend voor de volgende reacties:18Ne(α,p)21Na, 21Na(p,γ)22Mg, 22Mg(α,p)25Al, en 25Al(p,γ)26Si. De eerste reactie kaneen rol spelen bij het uitbreken uit de CNO cyclus, waardoor22Na en26Al kunnen wordengevormd.

Het onderzoek naar de kernstructuur van22Mg en 26Si is uitgevoerd met behulp vankernreacties, waarbij protonen als projectiel op de stabiele kernen24Mg, respectievelijk28Si, werden geschoten en waarbij tritium, dat tijdens deze botsing gevormd kan worden,is waargenomen. Deze kernreacties worden aangeduid als24Mg(p,t)22Mg en28Si(p,t)26Si.Het experimentele werk is uitgevoerd met behulp van de zogenaamde Grand Raiden mag-netische spectrometer van het Research Center of Nuclear Physics (RCNP) in Osaka. Degebruikte protonen werden daar versneld tot een energie van100 MeV met het Ring Cy-clotron; de tritium kernen werden waargenomen met de detectoren in het brandvlak vandeze magnetische spectrometer. Door gebruik te maken van een nieuwe bundellijn, spe-ciaal ontworpen voor experimenten waarmee een zeer goede energie resolutie verkregenkan worden, hebben we een energie resolutie kunnen bereikenvan 13 keV (volle breedteop halve hoogte van de piek). Een dergelijke resolutie was nog niet eerder bereikt voorvergelijkbare (p,t) reacties met de kernen22Mg en 26Si, die in het verleden waren uitge-voerd. Door deze zeer goede energie resolutie hebben we 62 energie niveaus in22Mg en 38energie niveaus in26Si kunnen identificeren. Van deze niveaus hebben we er 12 in22Mg,respectievelijk 14 in26Si, voor de eerste keer kunnen waarnemen en oplossen. De meestevan deze niveaus liggen in een energie gebied, dat belangrijk is voor astrofysische proces-sen. Het belang van een goede energieresolutie kan beoordeeld worden aan ons gemetenenergie spectrum van26Si, waar wij duidelijk in staat bleken te zijn de energieniveaus tescheiden die zich bevinden bij een excitatie energie van 4805.6 (± 2.5) keV en 4827.0 (±2.5) keV. Deze niveaus konden tot nu toe niet van elkaar onderscheiden waargenomen wor-den en werden daardoor als een niveau behandeld. Bovendien werd dit ene niveau tot nutoe gebruikt als ijkpunt voor de bepaling van de excitatie energie van niveaus bij een noghogere excitatie energie.

133

De gemeten energie spectra verkregen voor de24Mg(p,t)22Mg en28Si(p,t)26Si reactieszijn geijkt met behulp van de gepubliceerde data van referentie [16]. Deze onderzoekershebben voor de laagste energie niveaus in22Mg de excitatie energie zeer nauwkeurig be-paald met behulp vanγ-spectrometrie. Met deze gepubliceerde gegevens voor de energieniveaus in22Mg beneden de drempel, waarbij spontane emissie van protonen kan gaan op-treden, hebben we een systematische fout kunnen vermijden,die in de gegevens van Endt[46] zit. In ons onderzoek hebben we de onnauwkeurigheid in de bepaling van de exci-tatie energie van de niveaus kunnen verkleinen tot 1 keV in22Mg en respectievelijk tot 2keV in 26Si. In het bijzonder vermelden wij, dat vanwege deze hoge nauwkeurigheid deonzekerheid in de berekening van de astrofysische snelheidvoor de18Ne(α,p)21Na reactieverkleind kan worden tot 15%.

Andere belangrijke gegevens, die nodig zijn voor de berekening van de astrofysischereactiesnelheden zijn de waardes van de spin, de pariteit, and resonantie sterkte van de be-trokken energie niveaus. Tot nu toe zijn er slechts twee directe metingen gepubliceerd vande resonantiesterkte van aangeslagen toestanden in22Mg. Beide metingen zijn gebaseerdop studies, die uitgevoerd zijn met behulp van radioactievebundels. De eerste studie betreftonderzoek bij TRIUMF (Canada) voor de21Na(p,γ)22Mg reactie [17, 18, 19]. De twee-de studie is uitgevoerd in Louvain-la-Neuve (Belgie), waarbij de reactie18Ne(α,p)21Nagebruikt werd. Zoals uitgelegd is in paragraaf 5.1 zijn wij niet in staat geweest om een-duidige waarden toe te kennen voor de spin en de pariteit vooraangeslagen toestanden in22Mg en26Si. De ontbrekende gegevens hebben wij afgeleid uit de gegevens van spiegel-kernen, in dit geval22Ne, respectievelijk26Mg. We moeten hierbij vermelden, dat doordeze methode een inherente onzekerheid bestaat in de door ons berekende reactiesnelhe-den. Desalniettemin maken we door onze analyse een belangrijke stap voorwaarts in deberekening van de betrokken astrofysisch reactiesnelheden.

Met de nauwkeurige bepaling van de excitatie energie van toestanden in22Mg en26Sihebben we de onzekerheid in de berekende reactiesnelheden kunnen verkleinen. Bovendienlevert onze analyse belangrijke gegevens op voor de ijking van de excitatie energie vanniveaus, die bevolkt kunnen worden door andere reacties, die uitgevoerd kunnen wordenmet behulp van versnelde radioactieve kernen. Het onderliggende mechanisme van de(p,t) reactie bij de relatief hoge bundelenergie, die we gebruikt hebben, betekent dat wevoornamelijk toestanden bevolkt hebben met een zogenaamdenatuurlijke pariteit. Daaromis het noodzakelijk om ook kernreacties uit te voeren, waarbij toestanden bevolkt kunnenworden met een niet-natuurlijke pariteit. Als voorbeeld van dergelijke reacties noemenwe 12C(16O,6He)22Mg en 29Si(3He,6He)26Si. Het belang van toestanden met een niet-natuurlijke pariteit blijkt uit de25Al(p,γ)26Mg reactie, waar de belangrijkste bijdrage aande astrofysische reactiesnelheid toegeschreven kan worden aan twee resonanties, die spinen pariteit 1+ en 3+ hebben en dus een niet-natuurlijke pariteit hebben.

Dit onderzoek levert de eerste experimentele resultaten voor de belangrijke reactie22Mg(α,p)25Al. Onze berekende snelheden voor de reacties21Na(p,γ)22Mg en25Al(p,γ)26Si

134 Nederlandse samenvatting en vooruitblik

zijn vergelijkbaar met resultaten, die reeds zijn gepubliceerd [18, 28, 29]. Deze overeen-komst kan toegeschreven worden aan het gebruik van dezelfdewaarden voor de protonspectroscopische factoren in alle berekeningen en omdat debepaalde waarden voor de be-trokken resonantie energieen vrijwel gelijk zijn in alle uitgevoerde berekeningen. Voor de18Ne(α,p)21Na daarentegen is de berekende reactiesnelheid een factor 5groter dan eerderegepubliceerde resultaten[12, 20, 21, 64]. Het verschil kanverklaard worden, omdat in dehuidige studie nieuwe niveaus zijn gevonden boven de drempel waarbijα-deeltjes spontaanuitgezonden kunnen worden van22Mg.

De directe bepaling van de astrofysische snelheid voor de reacties18Ne(α,p)21Na,22Mg(α,p)25Al, en 25Al(p,γ)26Si bij een excitatie energie tot 2 MeV boven de drempel,waarbijα-deeltjes spontaan uitgezonden kunnen worden in respectievelijk 22Mg en26Si istot nu toe nog niet mogelijk gebleken, omdat intense bundelsvan de radioactieve kernen18Ne,22Mg, en25Al met een hoge kwaliteit nog niet geproduceerd kunnen worden. Daar-naast zijn voor dergelijke directe metingen ook detectiesystemen noodzakelijk, die in staatzijn α-deeltjes met een relatief lage kinetische energie waar te kunnen nemen.

Een methode, die gebruikt kan worden om de relatieve vertakkingverhouding voor hetverval van aangeslagen toestanden in niet-stabiele kernendoor emissie via protonen en viaα-deeltjes te bepalen is beschreven door Davidset al. [6, 15]. Deze zelfde methode zou ookgebruikt kunnen worden om het verval van aangeslagen toestanden in22Mg en26Si te be-studeren voor het energiegebied boven de drempel waarbij spontane emissie vanα-deeltjeskan optreden. Ook in dit geval kunnen deze toestanden bevolkt worden door (p,t) kernreac-ties uit te voeren in inverse reactie kinematica, waarbij zware kernen geschoten worden opzeer lichte kernen. De gemaakte tritium kernen en de terugstuitende zware kernen22Mg en26Si kunnen worden afgebogen en geanalyseerd met behulp van een enkele magnetischespectrometer. Zo kunnen bijvoorbeeld een aantal belangrijke producten voor de reactiewaarbij 24Mg op waterstof geschoten wordt in tijdsconcidentie gedetecteerd worden. Hetbetreft dan de kernen van tritium,21Na, en18Ne, die ontstaan via de24Mg(p,t)22Mg reac-tie, en waarbij vervolgens22Mg vervalt via proton-emissie, waardoor21Na gevormd wordtof via α-emissie, waardoor18Ne wordt gevormd.

Acknowledgment

Svega ovoga ne bi bilo da je Andrija juna 1989 godine otisao u frizere.I would like to mention here all the people who meant something to me personally

and professionally during my PhD study in Groningen over thelast four years. How-ever, I would first like to mention my supervisor Ivan Anicin, from the Faculty of Physics,Belgrade. Your incredible personality was decisive when I chose nuclear physics for mydiploma work and later for my PhD.

Second, I would like to thank to my current supervisor Ad van den Berg from KVI, firstfor the completely crazy decision to accept a PhD student from Serbia who was not able tospeak English (except the word HELP!). Furthermore, I wouldlike to honour you for yourgood guidance and advice during my work which brought me to a successful completionof my thesis. I must emphasize your patience with all my errors, and the really importantmotivation which you gave me. All my work would not have been finished on time if I didnot have the help and support at KVI from my promotor Muhsin Harakeh. Please acceptmy thanks for your ability to bring clarity every time I had doubts, and please accept myapology for the never ending variety of errors in my written English.

It would be inappropriate not to mention George Berg for his never ending enthusiasmand energy to perform the most unbelievable experiments in our collaboration. I mustalso emphasize his readiness to share his knowledge and experience with us. Michael- thank you for guidance in the field of nuclear astrophysics.I would like to mention hereother members of our collaboration: Kichiji Hatanaka, Yoshitaka Fujita, Shimbara, Adachi,Joachim, Paul, Shawn it was a pleasure to work with you.

Further, I would like to apologize to all the people at KVI whohad to enjoy the miseryof my loudness. And of course thanks for some memorable moments during brakes andFANTOM schools go to my fellow (“Americans”) colleagues at KVI: Mladen, Aran, Alma,

136 Acknowledgment

Peane, Ileana, Andrey, Emil, Shiva, Marko, Mirko, Igor, Marlene, Hossein (the first). Aran- thank you for help in every language (except Serbian and Russian) and for enduring mein the office.

Thanks also go to the Ex-Yugoslavian student comunity in Groningen: Diana, Mladene,Sasa, Mirela, Kaca, Sonja, Bore, Primoze, Janjo, Andreja, Gorane, Gorane, Graeme, Danilo,Branislava,Miro, Sretene, Danijela, whose friendship andpossibility for gossiping in mymother language helped me to transcend all difficult momentsduring my PhD.

I would like to mention the perfect volleyball Sundays whichgave me lot of pleasureand happiness, especially when Pavel lost the last game. Simone, Simon, Pavel, Roberto,Carol, Paola, Gilles, Patricio, Daniele, Cyril, Andrey, Dima, Branislava, Mladen, Stefania,Maria, Michele, Federika, Primoz.... Pity that we did not win the tournament. And I mustmention the nice nights which we spent together in the Pintelier.

Simon and Graeme thank you for helping with my English. (You will need to correctthis part also.)

There are many other people who I have possibly forgotten to mention and I apologizefor this. On other hand, I need to emphasize two people without whom my life in Groningenwould not have been that nice. Branislava - Thank you for everything, for all your help andall the nice funny moments within these four years, I am sorrythat I missed your weddingparty. Danijela Jovanovic hvala ti za sve sto si mi pruzila, izvini za sve.......

Of course I need to metion all my friends back home, for their support and the beau-tiful chats which I had with them during last four years. Maraja, Ljubice, Biljo, Ivane(izvini kume sto te pominjem na engleskom), Vlado, Stefane, Slavka, Dimice, Marija,Koki, Nensi, Tanja, Sanja, Natasa, Vlado, Nidzo thanks....

Ovde cu pomenuti moje mamu, tatu i sestru bez kojih ne bih uradio ovo sto sam uradio.Hvala vam za vasu bezrezervnu podrsku i razumevanje za svegluposti koje sam u zivoturadio. Od srca veliko hvala. I moram da naglasim Kaca u pravusi, ja nikada ne bih zavrsiofiziku da nije bilo tebe.

Na kraju moram da se zahvalim svim dosadasnjim (bez izuzetaka) vladama RepublikeSrbije sto sistematski gradanima ubijaju nadu za bolju buducnost i sto su mi olaksali odlukuza odlazak iz zemlje. Bez njihove “podrske” nikada ne bih doktorirao.

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